Question

Let $\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$ and $\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$ be two vectors Then which one of the following statements is TRUE ?

• A. Projection of $\vec{a}$ on $\vec{b}$ is $\frac{17}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec{b}$.
• B. Projection of $\vec{a}$ on $\vec{b}$ is $\frac{17}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec{b}$
• C. Projection of $\vec{a}$ on $\vec{b}$ is $\frac{-17}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec{b}$.
• D. Projection of $\vec{a}$ on $\vec{b}$ is $\frac{-17}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec{b}$.

Answer 1

The Correct Option is A

The correct option is(A): Projection of\(\vec{a}\) on \(\vec{b} \,\,is\) \(\frac{17}{\sqrt{35}}\) and the direction of the projection vector is same as of \(\vec{b}.\)

\(a=5\^{i}−\^{j}​−3\^{k}\)
\(b=\^{i}−3\^{j}​+5\^{k}\)
\(a⋅\^{b}=35​5−3−15​=−35​−13​\)

Answer 2

The projection of 𝒢 on 𝒲 is given by: 

Projection = 𝒢 · 𝒲 
Magnitude of 𝒲: ∥𝒲∥.

Calculate 𝒢 · 𝒲:

𝒢 · 𝒲 = (5)(1) + (−1)(3) + (−3)(5).

𝒢 · 𝒲 = 5 − 3 − 15 = −13.

Magnitude of 𝒲:

\(∥𝒲∥ = \sqrt{1^2 + 3^2 + 5^2} = \sqrt{1 + 9 + 25} = \sqrt{35}.\)

Therefore, the projection of 𝒢 on 𝒲 is:

Projection = \(𝒢 · 𝒲 / ∥𝒲∥ = -13 / \sqrt{35}.\)

The negative sign indicates that the projection vector is opposite in direction to 𝒲.

Conclusive Answer:

The correct answer should match the projection value \(\frac{-13}{\sqrt{35}},\) but this question is marked as dropped (DROP) since no option matches the calculation.