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 ?
Projection of $\vec{a}$ on $\vec{b}$ is $\frac{-13}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec{b}$.
- 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 $\vec{b}$
- Projection of $\vec{a}$ on $\vec{b}$ is $\frac{-17}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec{b}$.
- 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 $\vec{b}$.
The Correct Option is A
Approach Solution - 1
The correct option is(A): Projection of\(\vec{a}\) on \(\vec{b} \,\,is\) \(\frac{-13}{\sqrt{35}}\) and the direction of the projection vector is same as of \(\vec{b}.\)
\(a=5\^{i}−\^{j}−3\^{k}\)
\(b=\^{i}−3\^{j}+5\^{k}\)
\(a⋅\^{b}=355−3−15=−35−13\)
Approach Solution -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.
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A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail. It is denoted as
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