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Addition of Vectors

The vector ...

Question

The vector $b = 3 j + 4 k$ is to be written as the sum of a vector $b_{1}$ parallel to $a = i + j$ and a vector $b_{2}$ perpendicular to $a$ . Then $b_{1}$ is equal to

\frac{3}{2} (i + j)

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\frac{2}{3} (i + j)

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\frac{1}{2} (i + j)

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\frac{1}{3} (i + j)

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Solution

The correct option is C $\frac{3}{2} (i + j)$
$b_{1} ∥ a \Rightarrow b_{1} = a (i + j)$
$b_{2} = b - b_{1} = (3 - a) i - a j + 4 k$
Also, $b_{3} - a = 0$
$\Rightarrow (3 - a) - a = 0 \Rightarrow a = \frac{3}{2}$
$∴ b_{1} = \frac{3}{2} (i + j)$

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