Question

Find the value of ‘m’ for which the vectors \[a = 2i + mj - 3k\] and \[b = i - 2j + k\] are perpendicular?

Answer 1

Hint: We have a simple problem. we know that if two vectors are perpendicular then their dot product is zero. That is if vector ‘a’ and vector ‘b’ are perpendicular then \[ a.b = 0\]. Using this concept we can find the value of m.

Complete step by step solution:
Given,
\[a = 2i + mj - 3k\] and \[b = i - 2j + k\].
These two vectors are perpendicular,
 \[ \Rightarrow a.b = 0\]
\[ \Rightarrow \left( {2i + mj - 3k} \right).\left( {i - 2j + k} \right) = 0\]
Applying dot product we have,
\[ \Rightarrow 2 - 2m - 3 = 0\]
This is because we know that \[i.i = j.j = k.k = 0\].
\[ \Rightarrow - 2m = 3 - 2\]
\[ \Rightarrow - 2m = 1\]
Divide by -2 on both sides we have,
\[ \Rightarrow m = - \dfrac{1}{2}\]. This is the required answer.

Thus the required answer is \[m = - \dfrac{1}{2}\].

Note: The addition of vectors: the addition of the two vectors is done by adding the corresponding elements of two vectors. That is \[ \Rightarrow V\left( {a + b} \right) = V\left( a \right) - V\left( b \right)\].

Scalar multiplication: a scalar product of a vector is done by multiplying the scalar product with each of its terms individually. That is \[ \Rightarrow V\left( {s \times a} \right) = s \times V\left( a \right)\]