find the scalar and vector projection of 3`hat i - hat j + 4hat k
on 2 hat i + 3 hat j -6 hat k`

Explanation

To find the scalar and vector projections of the vector \( \mathbf{A} = 3\hat{i} - \hat{j} + 4\hat{k} \) onto the vector \( \mathbf{B} = 2\hat{i} + 3\hat{j} - 6\hat{k} \), we will follow these steps:

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

The dot product of two vectors \( \mathbf{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \mathbf{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \]
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