Question

The two vectors \[\overrightarrow{A}=\widehat{i}+2\widehat{j}+2\widehat{k}\] and \[\overrightarrow{B}=\widehat{i}-\widehat{j}+n\widehat{k}\] are perpendicular to each other, find the value of n.

Answer 1

Hint: Vector quantities are such quantities which have both magnitude and direction. On the other hand, quantities which have only magnitude are called scalar quantities. Vectors can be multiplied with one other using two product rules- broadly dot product which gives a scalar result and cross-product which gives vector result.

Step by step answer: We have two vectors \[\overrightarrow{A}\And \overrightarrow{B}\]
Dot product or scalar product of two vectors is given by \[\overrightarrow{A}.\overrightarrow{B}=AB\cos \alpha \]
Where \[\alpha \]is the angle between the two vectors. Now it is given they are perpendicular and we know if two vectors are perpendicular to each other then their dot product is zero.
Thus, \[\overrightarrow{A}.\overrightarrow{B}=0\]
\[(\widehat{i}+2\widehat{j}+2\widehat{k}).(\widehat{i}-\widehat{j}+n\widehat{k})=0\]
\[
\Rightarrow 1-2+2n=0 \\
 \Rightarrow 2n=1 \\
 \therefore n=0.5 \\
\]
Thus, the value of n comes out to be 0.5.

Additional information:
\[\overrightarrow{A}.\overrightarrow{B}=AB\cos \alpha \]
Let us look at how the value of cos \[\alpha \] varies over the given interval. We know the domain of cos is from -1 to 1. It attains its minimum value, -1 and maximum value, +1. Also, the value of cos \[\alpha \]is equal to zero when \[\alpha \]is equal to \[90{}^\circ \]. And when the value of the angle between the two vectors is \[90{}^\circ \] then the two vectors are said to be perpendicular to each other. - There are two laws to add the vectors, they are triangle law of vector addition and parallelogram law of vector addition.

Note: While taking either dot product or cross product we have to keep in mind we have to take the angle between the two original vectors. Also, while finding out the dot product, we multiply the component of one vector with another only.