Question

Find ‘m’ if \[\vec P = \hat i - 3\hat j + 4\hat k\] and \[\vec Q = m\hat i - 6\hat j + 8\hat k\] have the same direction.

Answer 1

Hint: Use the condition for having two vectors in the same direction. This condition states that the one vector is the scalar multiple of the other vector. The components of unit vectors of two vectors on either side of the equal sign must be the same. Using this statement determines the value of the scalar and then the value of m given in the vector.

Formula used:
The condition for having two vectors in the same direction is
\[\vec P = k\vec Q\] …… (1)
Here, \[\vec P\] and \[\vec Q\] are the two vectors in the same direction and \[k\] is the scalar.

Complete step by step answer:
We have given that the two vectors \[\vec P\] and \[\vec Q\] as
\[\vec P = \hat i - 3\hat j + 4\hat k\]
\[\vec Q = m\hat i - 6\hat j + 8\hat k\]

We have given that the two vectors \[\vec P\] and \[\vec Q\] are in the same direction. So these two vectors must satisfy the condition given in equation (1).
Substitute \[\hat i - 3\hat j + 4\hat k\] for \[\vec P\] and \[m\hat i - 6\hat j + 8\hat k\] for \[\vec Q\] in equation (1).
\[\left( {\hat i - 3\hat j + 4\hat k} \right) = k\left( {m\hat i - 6\hat j + 8\hat k} \right)\]
\[ \Rightarrow \hat i - 3\hat j + 4\hat k = km\hat i - 6k\hat j + 8k\hat k\]

The components of three unit vectors on either side of the equal sign in the above equation must be the same. Hence, we get three equations as
\[ \Rightarrow km = 1\] …… (2)
And
\[6k = 3\]
\[ \Rightarrow k = \dfrac{1}{2}\]
And
\[8k = 4\]
\[ \Rightarrow k = \dfrac{1}{2}\]

Substitute \[\dfrac{1}{2}\] for \[k\] in equation (2).
\[ \Rightarrow \dfrac{1}{2}m = 1\]
\[ \therefore m = 2\]

Hence, the value of m is 2.

Note:One can also solve the same question by another method. One can also use the information that the angle between the two vectors in the same direction is zero. Hence, use the formula for the dot product of the two given vectors and determine the value of m using the angle between the two vectors as zero. One can also directly equate the components of unit vectors of the given vectors in the same direction to determine the value of m in the vector Q.