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Mathematics · 2026 · 2 marks
CBSE 2026 · Region 1 · Set 1 · Q23
Find the vector of magnitude 14 in the direction of QP, where P and Q are the points (1,3,2) and (-1,0,8) respectively.
Find the vector of magnitude $\displaystyle 14$ in the direction of $\displaystyle \overrightarrow{\mathrm{QP}}$, where P and Q are the points $\displaystyle (1,3,2)$ and $\displaystyle (-1,0,8)$ respectively.
Official answer
($\displaystyle 8$, $\displaystyle 6$, -$\displaystyle 4$)
Marking-scheme solution
$\displaystyle \vec{QP} = 2\hat i + 3\hat j - 6\hat k$
$\displaystyle |\vec{QP}| = \sqrt{4 + 9 + 36} = 7$
The unit vector in the direction of $\displaystyle \vec{QP} = \dfrac{\vec{QP}}{|\vec{QP}|} = \dfrac{2}{7}\hat i + \dfrac{3}{7}\hat j - \dfrac{6}{7}\hat k$
The required vector $\displaystyle = 14\left(\dfrac{\vec{QP}}{|\vec{QP}|}\right) = 4\hat i + 6\hat j - 12\hat k$
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