Question

If the vectors $\overrightarrow {\text{k}}$ and $\overrightarrow {\text{A}}$ are parallel to each other ,then what is ${\text{k}}\overrightarrow {\text{k}} \times \overrightarrow {\text{A}}$ equal to?
A) ${k^2}\overrightarrow A$
B) $\overrightarrow 0$
C) $- {k^2}\overrightarrow A$
D) $\overrightarrow A$

Answer 1

Hint: Here since the vectors $\overrightarrow {\text{k}}$ and $\overrightarrow {\text{A}}$ are parallel to each other therefore the angle between them is . Hence apply the formula for $|\overrightarrow X \times \overrightarrow {Y|}$ to get the answer.
Formula of magnitude is given by:
$|\overrightarrow {\text{X}} \times \overrightarrow {{\text{Y}}|} = {\text{|}}\overrightarrow {\text{X}} {\text{| |}}\overrightarrow {{\text{Y|}}} \sin \theta$

Complete step by step solution:
We are given two vectors $\overrightarrow {\text{k}}$ and $\overrightarrow {\text{A}}$ such that they are parallel to each other.
Hence the angle between these two vectors is ${0^ \circ }$.
According to the following formula of vector product of two vectors:
$|\overrightarrow {\text{X}} \times \overrightarrow {{\text{Y}}|} = {\text{|}}\overrightarrow {\text{X}} {\text{| |}}\overrightarrow {{\text{Y|}}} \sin \theta$
Applying this formula for given vectors we get:
$\overrightarrow {{\text{|k}}} \times \overrightarrow {\text{A}} | = \overrightarrow {{\text{|k}}} |\overrightarrow {{\text{|A|}}} \sin {0^ \circ }$
Since , $\sin {0^ \circ } = 0$
Therefore , $\overrightarrow {{\text{|k}}} \times \overrightarrow {\text{A}} | = 0$
Hence,

$${\text{k}}\overrightarrow {\text{k}} \times \overrightarrow {\text{A}} = {\text{k}}\left( 0 \right) \\\ {\text{k}}\overrightarrow {\text{k}} \times \overrightarrow {\text{A}} = \overrightarrow 0 \\\$$

Therefore option B is the correct answer.

Note:
This question can be directly solved in one step as the vector product of all the vectors which are parallel to each other is zero.
Therefore,

$${\text{k}}\overrightarrow {\text{k}} \times \overrightarrow {\text{A}} = {\text{k}}\left( 0 \right) \\\ {\text{k}}\overrightarrow {\text{k}} \times \overrightarrow {\text{A}} = \overrightarrow 0 \\\$$