A vector $\vec{a}$ makes equal acute angles on the coordinate axis. Then the projection of vector $\vec{ b }=5 \hat{ i }+7 \hat{ j }-\hat{ k }$ on $\vec{ a }$ is
- $\frac{11}{15}$
- $\frac{11}{\sqrt{3}}$
- $\frac{4}{5}$
- $\frac{3}{5 \sqrt{3}}$
The Correct Option is B
Solution and Explanation
To find the projection of vector \(\vec{b} = 5 \hat{i} + 7 \hat{j} - \hat{k}\) on a vector \(\vec{a}\) that makes equal acute angles with the coordinate axes, we need to determine the direction and magnitude of \(\vec{a}\).
Since vector \(\vec{a}\) makes equal angles with the coordinate axes, each angle will be \(\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)\). Therefore, \(\vec{a}\) can be represented as:
\(\vec{a} = \frac{1}{\sqrt{3}} \hat{i} + \frac{1}{\sqrt{3}} \hat{j} + \frac{1}{\sqrt{3}} \hat{k}\)
The projection of a vector \(\vec{b}\) on another vector \(\vec{a}\) is given by the formula:
\(\text{Projection of } \vec{b} \text{ on } \vec{a} = \frac{\vec{b} \cdot \vec{a}}{\|\vec{a}\|}\)
First, calculate the dot product \(\vec{b} \cdot \vec{a}\):
\(\vec{b} \cdot \vec{a} = (5 \hat{i} + 7 \hat{j} - \hat{k}) \cdot \left(\frac{1}{\sqrt{3}} \hat{i} + \frac{1}{\sqrt{3}} \hat{j} + \frac{1}{\sqrt{3}} \hat{k}\right)\)
This results in:
\(= 5 \cdot \frac{1}{\sqrt{3}} + 7 \cdot \frac{1}{\sqrt{3}} + (-1) \cdot \frac{1}{\sqrt{3}}\)
\(= \frac{5 + 7 - 1}{\sqrt{3}} = \frac{11}{\sqrt{3}}\)
Next, calculate the magnitude of \(\vec{a}\):
\(\|\vec{a}\| = \sqrt{\left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{1}{\sqrt{3}}\right)^2}\)
\(= \sqrt{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = \sqrt{1} = 1\)
Thus, the projection of \(\vec{b}\) on \(\vec{a}\) is:
\(\text{Projection of } \vec{b} \text{ on } \vec{a} = \frac{11}{\sqrt{3}}\)
The correct answer is
\(\frac{11}{\sqrt{3}}\)
, which matches the given correct answer.
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Concepts Used:
Vector Algebra
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
The magnitude of the vector is represented as |V|. Two vectors are said to be equal if they have equal magnitudes and equal direction.
Vector Algebra Operations:
Arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.