Find the projection of the vector `7hat(i)+hat(j)-4hat(k)" on " 2hat(i)+6hat(j)+3hat(k)`. a) `8/7` b) `7/8` c) `6/7` d) `4/7`

Options

A. `8/7`
B. `7/8`
C. `6/7`
D. `4/7`

Answer

To find the projection of the vector \( \mathbf{A} = 7\hat{i} + \hat{j} - 4\hat{k} \) on the vector \( \mathbf{B} = 2\hat{i} + 6\hat{j} + 3\hat{k} \), we can follow these steps:

### Step 1: Calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \)

The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = (7)(2) + (1)(6) + (-4)(3) \]

Calculating each term:
- \( 7 \cdot 2 = 14 \)
- \( 1 \cdot 6 = 6 \)
- \( -4 \cdot 3 = -12 \)

Now, summing these results:
\[ \mathbf{A} \cdot \mathbf{B} = 14 + 6 - 12 = 8 \]

### Step 2: Calculate the magnitude of vector \( \mathbf{B} \)

The magnitude of a vector \( \mathbf{B} \) is given by:
\[ |\mathbf{B}| = \sqrt{(2)^2 + (6)^2 + (3)^2} \]

Calculating each term:
- \( (2)^2 = 4 \)
- \( (6)^2 = 36 \)
- \( (3)^2 = 9 \)

Now, summing these results:
\[ |\mathbf{B}| = \sqrt{4 + 36 + 9} = \sqrt{49} = 7 \]

### Step 3: Calculate the projection of \( \mathbf{A} \) on \( \mathbf{B} \)

The formula for the projection of vector \( \mathbf{A} \) on vector \( \mathbf{B} \) is given by:
\[ \text{proj}_{\mathbf{B}} \mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|^2} \mathbf{B} \]

First, we need \( |\mathbf{B}|^2 \):
\[ |\mathbf{B}|^2 = 7^2 = 49 \]

Now substituting the values into the projection formula:
\[ \text{proj}_{\mathbf{B}} \mathbf{A} = \frac{8}{49} \mathbf{B} \]

### Step 4: Find the scalar projection

The scalar projection of \( \mathbf{A} \) on \( \mathbf{B} \) is given by:
\[ \text{Scalar Projection} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} \]
Substituting the known values:
\[ \text{Scalar Projection} = \frac{8}{7} \]

### Final Answer

Thus, the projection of the vector \( \mathbf{A} \) on \( \mathbf{B} \) is \( \frac{8}{7} \).