Question

What is the magnitude of the projection of the vector $\overrightarrow r = 3\hat i + \hat j + 2\hat k$ on the x-y plane?
A. 3
B. 4
C. $\sqrt {14}$
D. $\sqrt {10}$

Answer 1

Hint
The projection of any n-dimensional quantity can be thought of as its shadow. As our shadows reduce to 2D, so would the projection of a 3-dimensional vector.
Formula used: $m = \sqrt {{a^2} + {b^2} + {c^2}}$ , where $a$ , $b$ and $c$ are the values in the X, Y and Z directions, respectively.

Complete step by step answer
We are provided with a vector that has a projection on the X-Y plane. We have to calculate the magnitude of this projection. Now, we know that the projection on any plane implies that the third dimension has been reduced to zero.
We are provided with the following 3-dimensional vector:
$\overrightarrow r = 3\hat i + \hat j + 2\hat k$
As we are told that the vector projects on the X-Y plane, it implies that its z-coordinate has become null. Putting z as zero would give us the projection vector as:
$\overrightarrow r = 3\hat i + \hat j$ [Here, the $\hat k$ refers to the Z direction]
We also know that the magnitude of a vector is calculated as:
$m = \sqrt {{a^2} + {b^2} + {c^2}}$
According to the calculated projection of the vector:
Value in X-direction $a = 3$
Value in Y-direction $b = 1$
Putting this in the equation to calculate the magnitude:
$m = \sqrt {{3^2} + {1^2}}$
$\Rightarrow m = \sqrt {9 + 1} = \sqrt {10}$
Hence, the magnitude of projection is $\sqrt {10}$ .
Therefore, the correct answer is option (D).

Note
Projections of vectors are helpful in determining their component in a particular direction on a plane. For example, consider a vector having X and Y components. We do not know what the vector would like solely from the X-axis unless we take a projection of it. This is used to simplify calculations and identify peculiar properties in a certain direction.