Vector Algebra
Projection of a vector on another; direction
nta_pyq_2023_jan
Grade 12
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?
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 $\vec{b}$
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 $\vec{b}$
Step-by-Step Solution
Key Concept: Projection of $\vec{a}$ on $\vec{b}$ = $\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$. Sign determines direction.
Projection $= \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} = \frac{17}{\sqrt{35}}$ (positive), direction same as $\vec{b}$. Answer: (1)
Correct Answer: Projection of $\vec{a}$ on $\vec{b}$ is $\frac{17}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec{b}$