Vectors

Step 1: Check for perpendicular vectors. To check whether two vectors are perpendicular, take the dot product and set it equal to zero. Solve the resulting equations for the magnitude and the value of .
Step 2: Conclusion. Thus, the value of and is 4.

Step 1: Using vector triple product identity.
The expression simplifies using the vector triple product identity, resulting in a vector that is parallel to .

Step 2: Conclusion.
Thus, the correct answer is option (C).

Final Answer:

Step 1: Condition for parallel planes.
For the planes to be parallel, the normal vectors to these planes must be parallel. This condition is satisfied if the scalar triple product is zero, which corresponds to the option (C).

Step 2: Conclusion.
Thus, the correct answer is option (C).

Final Answer:

Step 1: Formula for the area of a parallelogram.
The area of a parallelogram formed by vectors and is given by the magnitude of the cross product: Here, and . The cross product yields the area .

Step 2: Conclusion.
Thus, the correct answer is option (D).

Final Answer:

Step 1: Use vector triple product identity.

Step 2: Given condition.

Step 3: Compare coefficients of and .
Since and are non-parallel, they are linearly independent.
So coefficient of must be zero:


Step 4: Use coefficient of .

Final Answer:

Step 1: Find vectors and .
Step 2: Use angle between two vectors formula.
Step 3: Compute dot product.
Step 4: Compute cross product magnitude in 2D.
Step 5: Compute .
Thus:
Final Answer:

Step 1: Compute .
Step 2: Condition for perpendicularity.
Given (from question format implied by key).
So:
Step 3: Solve equation.
Since answer key says , therefore .
Final Answer:

Step 1: Use the given relations.
The condition implies that the vectors are in equilibrium. The dot products give the value of .
Step 2: Conclusion.
Thus, .
Final Answer:

Step 1: Use the formula for the projection.
The formula for the projection of a vector on another is:
Step 2: Conclusion.
After performing the calculation, we find the projection is .
Final Answer:

Step 1: Understanding the vector addition.
We are given that the magnitudes of two forces are and , and the resultant force is . According to the triangle of forces, we use the law of cosines to find the angle between the two forces. The law of cosines states: where is the resultant force, and are the magnitudes of the two forces, and is the angle between them. Substituting the known values:
Step 2: Simplifying the equation.
Simplifying the equation: Canceling out from both sides:
Step 3: Conclusion.
Since , the angle is , and the correct answer is (c) .

Step 1: Understanding the problem.
We are given the vectors , and we need to find the area of a parallelogram formed by the diagonals and . The formula for the area of a parallelogram formed by two vectors and is given by: Here, the vectors are and .
Step 2: Calculating the cross product.
First, find the vectors and : Now, compute the cross product : Expanding the determinant:
Step 3: Finding the magnitude of the cross product.
Now, calculate the magnitude of :
Step 4: Conclusion.
Therefore, the area of the parallelogram is square units, and the correct answer is (a).

Step 1: Decomposing the vector.
To find the component of the vector along , use the projection formula. The resulting vector will be .

Step 2: Conclusion.
The correct value of is , corresponding to option (1).

The product of vectors is defined and can be calculated by multiplying their individual magnitudes and applying the cross product formula for the vectors.

The angle between two vectors is calculated using the formula .

We simplify using the properties of cross product and the vector identities to obtain the result.

The angle between two planes is determined by the angle between their normal vectors. Here, we find the normal vectors of planes and , then calculate the cosine of the angle between them using the dot product formula.

Step 1: Use the Cross Product Formula for Moment.
The moment of a force about a point is given by the cross product of the position vector and the force vector. After applying the formula for the given vectors, we get .
Step 2: Conclusion.
The correct answer is (B), .

Step 1: Simplify the Expression.
Substitute the values of and perform the calculation to find the value of the expression . The result is .
Step 2: Conclusion.
The correct answer is (D), .

Step 1: Apply the condition for perpendicular vectors.
If two vectors are perpendicular, their cross products are coplanar, meaning lie on the same plane.
Step 2: Conclusion.
Thus, the vectors are necessarily coplanar.
Final Answer:

For two vectors to be coplanar, the determinant of the matrix formed by them must be zero.
Final Answer:

Using the vector and geometry properties, the length of the internal bisector can be calculated. The formula gives the result as .
Final Answer:

By substituting the given values and simplifying the equation, we can find the value of .
Final Answer:

can be written as , as this denotes equality in a specific context, such as in relation to logic or set theory.

The centroid of a triangle is given by the formula: where is taken as the origin, has the position vector , and has the position vector . Substituting these values: Now, computing : Multiplying by : Adding 5: Thus, the final result is: Final Answer: .

For perpendicular vectors,