Question

What is the equivalent vector of \[\left( {a \cdot \widehat i} \right)\widehat i + \left( {a \cdot \widehat j} \right)\widehat j + \left( {a \cdot \widehat k} \right)\widehat k\]?
A. \[a\]
B. \[2a\]
C. \[3a\]
D. 0

Answer 1

Hint: First we will let a vector \[a\]. Then apply the dot product and simplify it.

Formula used:
If \[\overrightarrow A = {x_1}\widehat i + {y_1}\widehat j + {z_1}\widehat k\] and \[\overrightarrow B = {x_2}\widehat i + {y_2}\widehat j + {z_2}\widehat k\], then the dot product of \[\overrightarrow A \] and \[\overrightarrow B \] is \[{x_1}{x_2} + {y_1}{y_2} + {z_1}{z_2}\].

Complete step by step solution:
Let the vector \[a\] be \[x\widehat i + y\widehat j + z\widehat k\].
 We will substitute \[a = x\widehat i + y\widehat j + z\widehat k\] in the given vector \[\left( {a \cdot \widehat i} \right)\widehat i + \left( {a \cdot \widehat j} \right)\widehat j + \left( {a \cdot \widehat k} \right)\widehat k\].
\[\left( {a \cdot \widehat i} \right)\widehat i + \left( {a \cdot \widehat j} \right)\widehat j + \left( {a \cdot \widehat k} \right)\widehat k = \left( {\left( {x\widehat i + y\widehat j + z\widehat k} \right) \cdot \widehat i} \right)\widehat i + \left( {\left( {x\widehat i + y\widehat j + z\widehat k} \right) \cdot \widehat j} \right)\widehat j + \left( {\left( {x\widehat i + y\widehat j + z\widehat k} \right) \cdot \widehat k} \right)\widehat k\]
\[\begin{array}{l} \Rightarrow \left( {a \cdot \widehat i} \right)\widehat i + \left( {a \cdot \widehat j} \right)\widehat j + \left( {a \cdot \widehat k} \right)\widehat k\\ = \left( {\left( {x\widehat i + y\widehat j + z\widehat k} \right) \cdot \left( {\widehat i + 0 \cdot \widehat j + 0 \cdot \widehat k} \right)} \right)\widehat i + \left( {\left( {x\widehat i + y\widehat j + z\widehat k} \right) \cdot \left( {0 \cdot \widehat i + \widehat j + 0 \cdot \widehat k} \right)} \right)\widehat j + \left( {\left( {x\widehat i + y\widehat j + z\widehat k} \right) \cdot \left( {0 \cdot \widehat i + 0 \cdot \widehat j + \widehat k} \right)} \right)\widehat k\end{array}\]
Now apply the dot product.
\[ \Rightarrow \left( {a \cdot \widehat i} \right)\widehat i + \left( {a \cdot \widehat j} \right)\widehat j + \left( {a \cdot \widehat k} \right)\widehat k = x\widehat i + y\widehat j + z\widehat k\] ……(i)
We assumed that \[x\widehat i + y\widehat j + z\widehat k = a\].
Now substitute the value of \[x\widehat i + y\widehat j + z\widehat k\] in equation (i).
\[ \Rightarrow \left( {a \cdot \widehat i} \right)\widehat i + \left( {a \cdot \widehat j} \right)\widehat j + \left( {a \cdot \widehat k} \right)\widehat k = a\]
Hence option A is the correct option.

Note: Many students confused with triple product and \[\left( {a \cdot \widehat i} \right)\widehat i\], \[\left( {a \cdot \widehat j} \right)\widehat j\], and \[\left( {a \cdot \widehat k} \right)\widehat k\]. But \[\left( {a \cdot \widehat i} \right)\widehat i\], \[\left( {a \cdot \widehat j} \right)\widehat j\], and \[\left( {a \cdot \widehat k} \right)\widehat k\] are not triple product. There involves a dot product. So it is not a triple product.