Vector Algebra

Step 1: Recall projection formula.
The projection of vector on vector is given by:

Step 2: Identify vectors.

Step 3: Compute dot product.

Step 4: Apply formula.

Step 5: Conclusion.
The projection is , hence the correct answer is (D).

The properties of cube roots of unity are:

Expanding the determinant:

Using cofactor expansion along the first row:

Evaluating each minor and simplifying using properties of :

To prove is an equivalence relation, we verify the three properties:

1. Reflexive: For any , , which is a multiple of 5. Hence, .
2. Symmetric: If , then is a multiple of 5. Thus, , which is also a multiple of 5. Hence, .
3. Transitive: If and , then and are multiples of 5. Adding these gives , which is a multiple of 5. Hence, .

Since satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.

Matrix multiplication is defined as:

Calculating the elements:

The given matrix is:

The transpose of is:

From the condition , where is the identity matrix:

Adding the matrices:

Equating elements:

The value of satisfying in the principal range is:

Hence, the correct answer is B) .

The determinant of a 2×2 matrix is given by: Substituting the values: Hence, the correct answer is A) 16.

For the first term:

Taking the logarithm:

Differentiating:

Thus:

For the second term:

Taking the logarithm:

Differentiating:

Thus:

Finally:

To find , compute: where is the determinant of and is the adjugate matrix of .

Step 1: Compute

Simplify to find .

Step 2: Compute

Find the cofactor matrix of and transpose it to get .

Step 3: Compute

For : 1. Differentiate : Let , then . 2. Differentiate : Using : Thus:

The given system of equations can be written in matrix form as: where To solve for , we use: Finding and computing , we obtain:

The inverse of the product of two invertible matrices follows the rule: The order of inversion is reversed due to matrix multiplication properties.

Step1:Representthesystemofequationsinmatrixform. Thegivensystemcanbewrittenas: where Step2:Findthedeterminantof . Expandingalongthefirstrow: Simplifyeachminor: Substitutethesevalues: Step3:Computetheinverseof . Theinverseof isgivenby: where istheadjointof .Computethecofactorsof tofind . Step4:Solvefor . Using ,computethevaluesof , ,and .

The given system of equations can be written in matrix form as: Let: Then: Find using the formula: Compute , , and then to solve for .

Step 1: Understanding the Concept:
A vector perpendicular to two given vectors can be found using their cross product. A unit vector is a vector with a magnitude of 1. The process involves first finding the two vectors and , then finding their cross product, and finally normalizing the resulting vector to get the unit vector.
Step 2: Key Formula or Approach:
Let and .
The vector perpendicular to both and is given by their cross product, .
The unit vector in the direction of is .
Step 3: Detailed Explanation or Calculation:
First, calculate and .
Given: and .
Let and .
Now, find the cross product .
Next, find the magnitude of .
Finally, find the unit vector .
Step 4: Final Answer:
The unit vector perpendicular to both and is . Note that the opposite vector is also a valid answer.

Step 1: Understanding the Concept:
This expression involves the dot product ( ) and cross product ( ) of the standard orthogonal unit vectors . A key issue is that the expression as written attempts to add scalars ( and ) and a vector ( ), which is mathematically undefined. This strongly implies a typographical error in the original question paper.
Step 2: Key Formula or Approach:
The standard properties of unit vectors are:
Dot product of a unit vector with itself: , , .
Cross product of any vector with itself is the zero vector: . Therefore, .
Given the invalidity of adding a scalar to a vector, we must assume the most likely typo. The pattern of the first two terms suggests the third term was also intended to be a dot product. Thus, we will solve the problem by assuming the expression is .
Step 3: Detailed Calculation:
Assuming the corrected expression is .
Let's evaluate each term based on the properties above:



Now, substitute these values back into the corrected expression: Step 4: Final Answer:
The calculated value is 1.

Step 1: Understanding the Concept:
The angle between two non-zero vectors and can be found using the scalar (dot) product formula. The dot product relates the angle between the vectors to the product of their magnitudes.
Step 2: Key Formula or Approach:
The dot product of two vectors and is defined as: From this, we can find the angle using: For vectors and :



Step 3: Detailed Calculation:
Let the given vectors be: 1. Calculate the dot product : 2. Calculate the magnitude of : 3. Calculate the magnitude of : 4. Calculate : 5. Find the angle : Step 4: Final Answer:
The angle between the two vectors is .

Step 1: Understanding the Concept:
The projection of a vector onto a non-zero vector is the orthogonal projection of onto a straight line parallel to . This results in a new vector that is parallel to .
Step 2: Key Formula or Approach:
The vector projection of onto is given by the formula:
This formula requires us to compute the dot product and the squared magnitude of .
Step 3: Detailed Explanation:
We are given the vectors:
First, we calculate the dot product .
Next, we calculate the magnitude of , denoted as .
Now, we find the square of the magnitude, .
Finally, we substitute these values into the projection formula.
Simplifying the scalar fraction: We can also distribute the scalar to get the vector components: Step 4: Final Answer:
The projection of vector on vector is .

Step 1: Understanding the Concept:
A unit vector is a vector with a magnitude (or length) of 1. To find the unit vector in the direction of a given vector, we divide the vector by its magnitude.
Step 2: Key Formula or Approach:
The formula for the unit vector in the direction of vector is: where is the magnitude of .
Step 3: Detailed Explanation:
The given vector is .
First, we calculate the magnitude of : Now, we divide the vector by its magnitude to get the unit vector : This can also be written as: Step 4: Final Answer:
The unit vector along is .

Step 1: Understanding the Concept:
The area of a parallelogram formed by two adjacent vectors and is given by the magnitude of their cross product, i.e., Area = .
Step 2: Key Formula or Approach:
1. Compute the cross product .
2. Calculate the magnitude of the resulting vector.
The cross product is calculated using the determinant of a matrix: Step 3: Detailed Explanation:
Given vectors: First, we find the cross product : Next, we find the magnitude of this new vector: Step 4: Final Answer:
The area of the parallelogram is square units.

Step 1: Understanding the Concept:
The dot product (or scalar product) of two vectors is a scalar quantity that relates the magnitudes of the vectors and the cosine of the angle between them. This relationship can be used to find the angle if the magnitudes and the dot product are known.
Step 2: Key Formula or Approach:
The formula for the dot product of two vectors and is: where and are the magnitudes (modulus) of the vectors and is the angle between them. We can rearrange this formula to solve for .
Step 3: Detailed Explanation:
We are given the following information:
Modulus of :
Modulus of :
Dot product:
We substitute these values into the dot product formula: Now, we solve for : Simplify the fraction: To rationalize the denominator, multiply the numerator and denominator by , or recognize that : Now, we find the angle whose cosine is . The principal value for the angle is: This is equivalent to 30°.
Step 4: Final Answer:
The angle between the vectors and is or 30°.

Step 1: Understanding the Concept:
The projection of a vector onto another vector is the scalar length of the component of that lies in the direction of .
Step 2: Key Formula or Approach:
The formula for the scalar projection of vector on vector is given by: We need to compute the dot product and the magnitude of .
Step 3: Detailed Explanation:
Given vectors are: First, calculate the dot product : Next, calculate the magnitude of : Now, substitute these values into the projection formula: Step 4: Final Answer:
The projection of the vector on the vector is .

Step 1: Understanding the Concept:
This problem requires proving a vector identity that relates the angle between two unit vectors to the magnitude of their difference. The proof will utilize the properties of the dot product and trigonometric half-angle identities.
Step 2: Key Formula or Approach:
We will use the following properties: 1. For any vector , its squared magnitude is . 2. The dot product of two vectors and is . 3. For unit vectors and , and . 4. The trigonometric half-angle identity: .
Step 3: Detailed Explanation:
Let's start by considering the expression .
Using the dot product property, we can write: Expanding the dot product: Since and are unit vectors, and .
The dot product is commutative, so . The expression becomes: Now, substitute the definition of the dot product: . Using the half-angle identity : Taking the square root of both sides: The angle between two vectors is usually taken in the range . Therefore, is in the range , where the sine function is non-negative. So, . Step 4: Final Answer:
Rearranging the equation, we get: This completes the proof.

Step 1: Understanding the Concept:
The area of a parallelogram can be determined using the vector cross product. When the diagonals of the parallelogram are given as vectors, there is a specific formula that relates the area to the magnitude of the cross product of these diagonal vectors.
Step 2: Key Formula or Approach:
If and are the vectors representing the diagonals of a parallelogram, the area of the parallelogram is given by: Step 3: Detailed Explanation:
Here, the diagonal vectors are and .
First, we need to compute the cross product . Next, we find the magnitude of this resulting vector: Finally, we use the formula for the area: Step 4: Final Answer:
The area of the parallelogram is square units.

Step 1: Understanding the Concept:
To find the vector connecting two points, say from point P to point Q, we subtract the position vector of the initial point (P) from the position vector of the terminal point (Q).
Step 2: Key Formula or Approach:
If P has coordinates and Q has coordinates , the vector is given by: This can also be written as .
Step 3: Detailed Explanation:
We are given the coordinates of the points:
P = (2, 3, 0), so .
Q = (–1, –2, –4), so .
Now, we apply the formula: Step 4: Final Answer:
The vector is . This matches option (C).

Step 1: Understanding the Concept:
This inequality is known as the triangle inequality for vectors. It states that the magnitude of the sum of two vectors is less than or equal to the sum of their individual magnitudes. Geometrically, it means that the length of any side of a triangle is less than or equal to the sum of the lengths of the other two sides. The proof relies on the properties of the dot product and the Cauchy-Schwarz inequality.
Step 2: Key Formula or Approach:
1. Start with the square of the magnitude, .
2. Use the property .
3. Expand the dot product .
4. Apply the Cauchy-Schwarz inequality, which states .
5. Show that .
6. Take the square root of both sides to obtain the final inequality.
Step 3: Detailed Explanation:
We consider the square of the magnitude of : Expanding the dot product: Using the properties and : From the definition of the dot product, we know , where is the angle between the vectors. The Cauchy-Schwarz inequality states that (since ).
Applying this to our expression: The right side of the inequality is the expansion of .
So, we have: Since magnitudes are non-negative, we can take the square root of both sides without changing the inequality: Step 4: Final Answer:
The triangle inequality for vectors, , has been proven. The equality holds when the vectors and are in the same direction ( ).

Step 1: Understanding the Concept:
The problem asks for the magnitude of the difference between two vectors, given their individual magnitudes and their dot product. The key is to relate the magnitude of a vector to the dot product of the vector with itself.
Step 2: Key Formula or Approach:
We will use the property that for any vector , its magnitude squared is given by .
1. Apply this property to the vector : .
2. Expand the dot product using its distributive property.
3. Substitute the given values of , , and .
4. Take the square root to find .
Step 3: Detailed Explanation or Calculation:
We want to find . Let's start by calculating its square: Expand the dot product: Using the properties and the commutative property of the dot product ( ): Now, substitute the given values: , , and . Now, take the square root of both sides. Since magnitude must be non-negative: Step 4: Final Answer:
The value of is .

Step 1: Understanding the Concept:
This problem involves the scalar triple product and the properties of the standard orthogonal unit vectors .
The expression is known as the scalar triple product.
Step 2: Key Formula or Approach:
We need to use the cyclic properties of the cross product of unit vectors:



And the anti-cyclic properties:



We also use the dot product properties: .
Step 3: Detailed Explanation:
Let's evaluate each term of the expression separately.
Term 1: Using the cyclic property, . Term 2: Using the anti-cyclic property, . Term 3: Using the cyclic property, . Now, add the values of the three terms: Step 4: Final Answer:
The value of the given expression is 1.

Step 1: Understanding the Concept:
This question checks the understanding of two different properties of vectors: magnitude and equality.
Magnitude of a vector: The length of the vector. For , the magnitude is .
Equality of vectors: Two vectors are equal if and only if all their corresponding components are identical.
Step 2: Key Formula or Approach:
1. Calculate and using the magnitude formula and compare them.
2. Compare the corresponding components of and to check for equality.
Step 3: Detailed Explanation or Calculation:
Given vectors: 1. Check if Magnitudes are Equal:
Calculate the magnitude of : Calculate the magnitude of : Since both magnitudes are equal to , we can conclude that .
2. Check if Vectors are Equal:
For two vectors to be equal, their corresponding components must be equal.
Vector can be written as .
Vector can be written as .
Comparing the components: .
Comparing the components: .
Since the components are not identical, the vectors and are not equal.
Step 4: Final Answer:
Yes, the magnitudes are equal: .
No, the vectors are not equal because their corresponding components are different.

Step 1: Understanding the Concept:
Direction cosines of a vector are the cosines of the angles that the vector makes with the positive x, y, and z axes. They are also the components of the unit vector in the direction of the given vector.
Step 2: Key Formula or Approach:
For a vector : 1. Find the magnitude of the vector: .
2. The direction cosines ( ) are given by: Step 3: Detailed Explanation:
The given vector is .
The components of the vector are , , and . These are the direction ratios.
1. Calculate the magnitude of : 2. Calculate the direction cosines: So, the direction cosines are .
Step 4: Final Answer:
The direction cosines of the vector are .

Step 1: Understanding the Concept:
This inequality is known as the triangle inequality for vectors. It states that the magnitude of the sum of two vectors is less than or equal to the sum of their individual magnitudes. We can prove it using the properties of the dot product.
Step 2: Key Approach:
It is easier to work with the squares of the magnitudes to avoid dealing with square roots. We will prove the equivalent inequality .
Step 3: Detailed Proof:
We start by considering the square of the magnitude of the sum of the vectors: Expanding the dot product using its distributive property: We know that and the dot product is commutative ( ). By the definition of the dot product, , where is the angle between the vectors.
We know that the maximum value of is 1. Therefore, we have the inequality: Substituting this into our expression: The right side of the inequality is the expansion of a perfect square: Since magnitude is a non-negative quantity, we can take the square root of both sides without changing the direction of the inequality: Step 4: Final Answer:
We have successfully proven the triangle inequality for vectors.

Step 1: Understanding the Concept:
The area of a triangle in three-dimensional space with given vertices can be found using the cross product of two vectors that represent adjacent sides of the triangle. The magnitude of this cross product is equal to the area of the parallelogram formed by these vectors, and the triangle's area is half of that.
Step 2: Key Formula or Approach:
The area of is given by the formula: The steps are: 1. Find the vectors for two adjacent sides, e.g., and . 2. Compute their cross product. 3. Calculate the magnitude of the resulting vector and divide by 2.
Step 3: Detailed Explanation:
The vertices are A(2, 2, 2), B(2, 1, 3), and C(3, 2, 1).
First, determine the vectors for two sides starting from vertex A: Next, calculate the cross product : Now, find the magnitude of this cross product vector: Finally, the area of the triangle is half of this magnitude: Step 4: Final Answer:
The area of the triangle is square units.

Step 1: Understanding the Concept:
To find the unit vector along the line AB, we first need to find the vector by subtracting the position vector of the initial point A from the position vector of the terminal point B. Then, we find the unit vector by dividing by its magnitude.
Step 2: Key Formula or Approach:
1. Let the position vectors of A and B be and respectively.
2. The vector along the line is .
3. The unit vector is .
Step 3: Detailed Explanation:
We are given the position vectors: First, calculate the vector : Next, calculate the magnitude of : Finally, find the unit vector by dividing by its magnitude: Step 4: Final Answer:
The unit vector along the straight line AB is .

Step 1: Understanding the Concept:
Direction cosines of a vector are the cosines of the angles the vector makes with the positive x, y, and z axes. For a vector , the direction cosines are , , and . We first need to find the sum of the given vectors.
Step 2: Key Formula or Approach:
1. Calculate the resultant vector .
2. Find the magnitude of the resultant vector, .
3. The direction cosines (l, m, n) are the components of the unit vector in the direction of .
Step 3: Detailed Explanation:
Given vectors are: First, find the sum : Next, find the magnitude of : Now, calculate the direction cosines: Step 4: Final Answer:
The direction-cosines of the sum of the vectors are .

Step 1: Understanding the Concept:
To show that three given position vectors form the vertices of a right-angled triangle, we can find the vectors representing the sides of the triangle.
If any two of these side vectors are perpendicular to each other, the triangle is a right-angled triangle.
Two vectors are perpendicular if their dot product (scalar product) is zero.
Step 2: Key Formula or Approach:
Let the position vectors of the vertices A, B, and C be , , and respectively.
The vectors representing the sides are , , and .
The condition for a right-angled triangle is that the dot product of any two side vectors is zero, for example, .
Step 3: Detailed Explanation or Calculation:
Let the given position vectors be:



Now, let's find the vectors for the sides of the triangle:






Next, we calculate the dot products of these side vectors:



Step 4: Final Answer:
Since the dot product , the sides represented by vectors and are perpendicular to each other.
Therefore, the triangle formed by the given position vectors is a right-angled triangle, with the right angle at vertex A.

Step 1: Understanding the Concept:
We can find the value of the expression involving dot products by using the given vector sum. The key is to take the dot product of the vector sum with itself, which introduces the square of the magnitudes and the dot product terms we are interested in.
Step 2: Key Formula or Approach:
- Given:
- Square this equation by taking the dot product with itself:
- Use the property .
Step 3: Detailed Explanation or Calculation:
Start with the given vector sum: Take the dot product of both sides with : Expand the left side: Using and the commutative property of the dot product ( ): Now, we can solve for the expression we want: Step 4: Final Answer:
The value of is .

Step 1: Understanding the Concept:
Two non-zero vectors are perpendicular if and only if their dot product (scalar product) is zero.
Step 2: Key Formula or Approach:
Let and .
If , then .
The dot product is calculated as: .
Step 3: Detailed Explanation or Calculation:
Let and .
Since the vectors are perpendicular, their dot product must be zero:
Step 4: Final Answer:
The value of is 6.

Step 1: Understanding the Concept:
The area of a parallelogram whose adjacent sides are represented by two vectors is equal to the magnitude of the cross product of those two vectors.
Step 2: Key Formula or Approach:
Area of parallelogram = .
The cross product is calculated using the determinant:
Step 3: Detailed Explanation or Calculation:
The given vectors are and .
First, we compute the cross product :
Expanding the determinant:
Next, we find the magnitude of this resulting vector:
Step 4: Final Answer:
The area of the parallelogram is square units.

Step 1: Understanding the Concept:
This problem uses the properties of the dot product of vectors, specifically that the dot product of a vector with itself gives the square of its magnitude. We can use the given condition to find the required value.
Step 2: Key Formula or Approach:
The key identity is . We will take the dot product of the vector sum with itself.
We start with the given equation:
Taking the dot product of both sides with :
Step 3: Detailed Explanation or Calculation:
Expanding the left side of the equation:
Using and the commutative property of the dot product ( ):
Now, substitute the given magnitudes: , , and .
Now, solve for the required expression:
Step 4: Final Answer:
The value of is .

Step 1: Understanding the Concept:
The inequality is known as the Cauchy-Schwarz inequality for vectors in Euclidean space. It relates the dot product of two vectors to the product of their magnitudes.
Step 2: Key Formula or Approach:
The proof relies on the geometric definition of the dot product (scalar product) of two vectors and :
where and are the magnitudes of the vectors, and is the angle between them.
Step 3: Detailed Explanation or Calculation:
We start with the definition of the dot product: Take the absolute value (or modulus) of both sides of the equation: Since the magnitudes and are non-negative scalars, we can take them out of the absolute value sign: Now, we consider the property of the cosine function. For any angle , the value of lies in the range .
This means that the absolute value of must be in the range .
Since , we can multiply both sides of this inequality by the non-negative quantity without changing the inequality direction: Substituting back in place of , we get: Step 4: Final Answer:
The inequality is proven based on the fundamental properties of the dot product and the cosine function. The equality holds when , which means or (i.e., when the vectors are collinear).

We need to evaluate the expression involving dot products and cross product of unit vectors.

Step 1: Recall the properties of unit vectors.
For the standard unit vectors :
-
-
- (cross product of a vector with itself is always the zero vector)

Step 2: Substitute these values into the expression.

Step 3: Simplify.

Step 4: Conclusion.
The value of the given expression is 0.
Final Answer: (A) 0