The projection of the vector `2hati+hatj -3hatk` on the vector `hati-2hatj-hatk` is

Options

A. `-3/(sqrt(14))`
B. `3/(sqrt(14))`
C. `-sqrt(3/2)`
D. `sqrt(3/2)`

Answer

To find the projection of the vector \( \mathbf{p} = 2\hat{i} + \hat{j} - 3\hat{k} \) on the vector \( \mathbf{q} = \hat{i} - 2\hat{j} - \hat{k} \), we can use the formula for the projection of vector \( \mathbf{p} \) onto vector \( \mathbf{q} \):

\[ \text{proj}_{\mathbf{q}} \mathbf{p} = \frac{\mathbf{p} \cdot \mathbf{q}}{\|\mathbf{q}\|^2} \mathbf{q} \]

### Step 1: Calculate the dot product \( \mathbf{p} \cdot \mathbf{q} \)

The dot product of two vectors \( \mathbf{p} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k} \) and \( \mathbf{q} = a_2 \hat{i} + b_2 \hat{j} + c_2 \hat{k} \) is given by:

\[ \mathbf{p} \cdot \mathbf{q} = a_1 a_2 + b_1 b_2 + c_1 c_2 \]

For our vectors:

- \( \mathbf{p} = 2\hat{i} + \hat{j} - 3\hat{k} \) (where \( a_1 = 2, b_1 = 1, c_1 = -3 \))
- \( \mathbf{q} = \hat{i} - 2\hat{j} - \hat{k} \) (where \( a_2 = 1, b_2 = -2, c_2 = -1 \))

Calculating the dot product:

\[ \mathbf{p} \cdot \mathbf{q} = (2)(1) + (1)(-2) + (-3)(-1) = 2 - 2 + 3 = 3 \]

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

The magnitude squared of a vector \( \mathbf{q} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by:

\[ \|\mathbf{q}\|^2 = a^2 + b^2 + c^2 \]

For \( \mathbf{q} \):

\[ \|\mathbf{q}\|^2 = (1)^2 + (-2)^2 + (-1)^2 = 1 + 4 + 1 = 6 \]

### Step 3: Substitute into the projection formula

Now we can substitute the values we calculated into the projection formula:

\[ \text{proj}_{\mathbf{q}} \mathbf{p} = \frac{\mathbf{p} \cdot \mathbf{q}}{\|\mathbf{q}\|^2} \mathbf{q} = \frac{3}{6} \mathbf{q} = \frac{1}{2} \mathbf{q} \]

### Step 4: Calculate the final projection vector

Now we multiply \( \frac{1}{2} \) by the vector \( \mathbf{q} \):

\[ \text{proj}_{\mathbf{q}} \mathbf{p} = \frac{1}{2} (\hat{i} - 2\hat{j} - \hat{k}) = \frac{1}{2} \hat{i} - \hat{j} - \frac{1}{2} \hat{k} \]

### Final Answer

The projection of the vector \( 2\hat{i} + \hat{j} - 3\hat{k} \) on the vector \( \hat{i} - 2\hat{j} - \hat{k} \) is:

\[ \frac{1}{2} \hat{i} - \hat{j} - \frac{1}{2} \hat{k} \]