Question
The projection of vector $\vec{A} = \hat{i}-2\hat{j}+\hat{k}$ on vector $\vec{B} = 4\hat{i}-4\hat{j}+7\hat{k}$ is:
The correct answer is
$19/9$
Projection of Vector A on Vector B
Given vectors:
\[ \vec{A} = \hat{i} - 2\hat{j} + \hat{k}, \quad \vec{B} = 4\hat{i} - 4\hat{j} + 7\hat{k} \]
Formula Used
Scalar Projection of A on B:
\[ \text{Projection} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} \]
Step 1: Dot Product
\[ \vec{A} \cdot \vec{B} = (1)(4) + (-2)(-4) + (1)(7) \]
\[ = 4 + 8 + 7 = 19 \]
Step 2: Magnitude of B
\[ |\vec{B}| = \sqrt{4^2 + (-4)^2 + 7^2} \]
\[ = \sqrt{16 + 16 + 49} = \sqrt{81} = 9 \]
Final Answer
\[ \text{Projection of } \vec{A} \text{ on } \vec{B} = \frac{19}{9} \]
Answer: 19/9
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