MHT CET · Maths · Three Dimensional Geometry
The vector projection of \(\bar{b}\) on, \(\bar{a}=3 \hat{i}+2 \hat{j}+5 \hat{k}\) and \(\bar{b}=7 \hat{i}-5 \hat{j}-\hat{k}\)
- A \(\frac{6(3 \hat{i}+2 \hat{j}+5 \hat{k})}{\sqrt{38}}\)
- B \(\frac{3(3 \hat{i}+2 \hat{j}+5 \hat{k})}{38}\)
- C \(\frac{3(3 \hat{i}+2 \hat{j}+5 \hat{k})}{19}\)
- D \(\frac{3(3 \hat{i}+2 \hat{j}+5 \hat{k})}{\sqrt{38}}\)
Answer & Solution
Correct Answer
(C) \(\frac{3(3 \hat{i}+2 \hat{j}+5 \hat{k})}{19}\)
Step-by-step Solution
Detailed explanation
The vector projection of \(\vec{b}\) on \(\vec{a}\)
\(\begin{aligned} & =\left(\frac{\vec{b} \cdot \vec{a}}{|\vec{a}|}\right) \hat{a} \\ & =\left(\frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2}\right) \vec{a} \\ & =\left(\frac{7 \times 3+(-5) \times 2+(-1) \times 5}{3^2+2^2+5^2}\right)(3 \hat{i}+2 \hat{j}+5 \hat{k}) \\ & =\frac{6}{38}(3 \hat{i}+2 \hat{j}+5 \hat{k}) \\ & =\frac{3}{19}(3 \hat{i}+2 \hat{j}+5 \hat{k})\end{aligned}\)
\(\begin{aligned} & =\left(\frac{\vec{b} \cdot \vec{a}}{|\vec{a}|}\right) \hat{a} \\ & =\left(\frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2}\right) \vec{a} \\ & =\left(\frac{7 \times 3+(-5) \times 2+(-1) \times 5}{3^2+2^2+5^2}\right)(3 \hat{i}+2 \hat{j}+5 \hat{k}) \\ & =\frac{6}{38}(3 \hat{i}+2 \hat{j}+5 \hat{k}) \\ & =\frac{3}{19}(3 \hat{i}+2 \hat{j}+5 \hat{k})\end{aligned}\)
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