Question

If a vector \[\left( {2\widehat i + 3\widehat j + 8\widehat k} \right)\] is perpendicular to the vector \[\left( {4\widehat i - 4\widehat j + a\widehat k} \right)\], then find the value of ‘a’.

Answer 1

Hint:Before we are going to solve this problem let’s see what data they have given and how to solve this problem. Here, they have given the two vectors in terms of their unit vectors. If both the vectors are perpendicular to each other, then we have to find the value of a. To solve this, first, we have to write the condition for the vectors that are perpendicular to one another and we obtain the value of a. so, now let’s solve this problem.

Formula Used:
The formula to find when the two vectors are perpendicular to each other,
\[\overrightarrow A .\overrightarrow B = 0\]……… (1)
Where, \[\overrightarrow A \] and \[\overrightarrow B \] are the two vectors.

Complete step by step solution:
Let \[\overrightarrow A = 2\widehat i + 3\widehat j + 8\widehat k\] and \[\overrightarrow B = 4\widehat i - 4\widehat j + a\widehat k\] are the two vectors and they are perpendicular to each other, that is, \[\overrightarrow A \bot \overrightarrow B \]. If \[\overrightarrow A \bot \overrightarrow B \] then, \[\overrightarrow A .\overrightarrow B = \left| A \right|\left| B \right|\cos \theta \]……. (2)

If vector \[\overrightarrow A \] and \[\overrightarrow B \] are perpendicular then, the angle between these two will be \[{90^0}\]. Then, equation (2) will become,
\[\overrightarrow A .\overrightarrow B = 0\] since, \[\cos {90^0} = 0\]
By using equation (1) we get,
\[\left( {2\widehat i + 3\widehat j + 8\widehat k} \right).\left( {4\widehat i - 4\widehat j + a\widehat k} \right) = 0\]
\[\Rightarrow 8 - 12 + 8a = 0\]
Because, \[\widehat i \cdot \widehat i = \widehat j \cdot \widehat j = \widehat k \cdot \widehat k = 1\]and \[\widehat i \cdot \widehat j = \widehat j \cdot \widehat k = \widehat k \cdot \widehat i = 0\]\[\]
\[ \therefore a = \dfrac{1}{2}\]

Therefore, the value of ‘a’ is \[\dfrac{1}{2}\].

Note:Unit vectors are defined as vectors which have a magnitude of 1. Any vector can be converted into a unit vector-only if it is divided by the magnitude of a given vector. To understand this, consider an example. A vector \[\overrightarrow A = \left( {2,3} \right)\] is considered, which has a magnitude of \[\left| A \right|\].If we divide each component of vector \[\overrightarrow A \] by \[\left| A \right|\] we will get the unit vector \[{u_A}\] that is in the same direction as \[\overrightarrow A \].