Vector Algebra

If and are unit vectors and is the angle between and , then we need to find .

Since and are unit vectors, and .

We know that .

.

Since and , we have:

We know that , so:

Taking the square root of both sides:

Therefore, the correct option is (C) .

The two vectors and represent the two sides and respectively of a .

We need to find the length of the median through A.

Let D be the midpoint of BC. The median through A is AD. We can find using the formula for the midpoint:

The length of the median is the magnitude of :

Therefore, the length of the median through A is .

Thus, the correct option is (D) .

Given that , it means and are perpendicular to each other. We are also told that makes an angle of 60° with . The angle between and is 60°. Using the formula for the dot product: Since , we get: Now calculate : because . The magnitude of is: Substitute these values into the dot product formula: Simplifying: Squaring both sides: Now, solving for and : Thus:

The correct answer is (D) : .

The area of a parallelogram formed by two vectors and is given by the magnitude of the cross product of these vectors: We are given that the area of the parallelogram with sides and is 15 square units, so: Now, we need to find the area of the parallelogram formed by the vectors and . The area of the new parallelogram is given by the magnitude of the cross product of these two vectors: We expand the cross product: Using the property that and , we get: Since , we get: Thus, the magnitude of this cross product is:

So, the correct answer is (C) : 105.

Given: Diagonals of a parallelogram are

Step 1: Sides of the parallelogram are half the sum and half the difference of the diagonals:

Step 2: Calculate

Step 3: Calculate

Step 4: Find magnitudes of the sides

Shorter side =

Correct option:

Since vector makes equal acute angles with the coordinate axes, we can conclude that its direction ratios are all equal. Thus, we assume that: The projection of vector on is given by the formula: First, we calculate the dot product : Next, we calculate the magnitude of : Now, we can find the projection: Thus, the projection of on is .

To find the projection of vector on a vector that makes equal acute angles with the coordinate axes, we need to determine the direction and magnitude of .

Since vector makes equal angles with the coordinate axes, each angle will be . Therefore, can be represented as:

The projection of a vector on another vector is given by the formula:

First, calculate the dot product :

This results in:

Next, calculate the magnitude of :

Thus, the projection of on is:

The correct answer is

, which matches the given correct answer.

To solve the problem, we start by interpreting the given conditions and applying relevant vector properties:

1. The condition specifies that vectors and are perpendicular.
2. Given that the angle between and is , we can utilize the cosine rule for vector angles:

For two vectors and , the cosine of the angle between them can be given by:

Setting the vectors as and , and the angle , we have:

Given , the equation becomes:

Since :

Solving for this, we have:

Simplifying this equation yields:

Taking the square root on both sides gives:

1. Thus, the correct answer is the option where .

Therefore, the correct answer is:

To find the area of the parallelogram having and as two adjacent sides, we need to use the properties of vector cross products to find the area formula.

1. The area of a parallelogram determined by vectors and is given by the magnitude of the cross product, .
2. We are given that the area of the parallelogram with sides and is 15, which implies: .
3. We need to find .
4. Using the distributive property of the cross product, we have: .
5. Since the cross product of any vector with itself is zero ( ), we simplify: .
6. Recall that , hence: .
7. Thus, the area of the new parallelogram is the absolute value: .

Therefore, the area of the parallelogram with sides and is 105 s units.

The correct answer is: (C): 6

We are given the following vector equations:

Step 1: Express in terms of and

From the first equation, solve for :

Step 2: Substitute into the second equation

Now, substitute into the second equation:

Step 3: Simplify the cross products

Now, simplify the terms involving the cross products. We know that , so and . Therefore, we have:

This simplifies to:

Step 4: Combine like terms

Now combine the terms involving and :

Since , substitute this into the equation:

Step 5: Factor the terms

Factor out on the left-hand side:

Step 6: Solve for

Now, cancel out (assuming ) to get:

Conclusion:
The value of is 6, so the correct answer is (C): 6.

The correct answer is: (D): 4

We are given the following information:

|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 144

|\vec{a}| = 4

We are tasked with finding .

Step 1: Use the identity for cross and dot products

We know the following identity for vectors and :

|\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2

Substitute this into the given equation:

(|\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2) + |\vec{a} \cdot \vec{b}|^2 = 144

Step 2: Simplify the equation

Notice that the second term and the last term involve , so they cancel out. This simplifies to:

|\vec{a}|^2 |\vec{b}|^2 = 144

Step 3: Substitute the known value for

We are given that , so substitute this into the equation:

4^2 |\vec{b}|^2 = 144

Step 4: Solve for

Now, simplify and solve for :

16 |\vec{b}|^2 = 144

|\vec{b}|^2 = \frac{144}{16} = 9

Taking the square root of both sides gives:

|\vec{b}| = 3

Conclusion:
The correct value of is , so the correct answer is (D): .

The correct answer is: (B): and are perpendicular.

We are given that:

Step 1: Square both sides

We begin by squaring both sides of the equation to eliminate the magnitudes:

Step 2: Expand both sides

Now, expand both sides using the formula for the square of the magnitude of a vector, :

Expanding both sides gives:

Step 3: Simplify the equation

Canceling the common terms and from both sides, we are left with:

Step 4: Solve for

By simplifying this, we get:

This implies:

Conclusion:
The dot product of and is zero, which means that the vectors and are perpendicular.