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  4. Given the vectors a=(3, -1,5) and b=(1,2,-3) . A vector c is such that it is perpendicular to the z-axis and satisfies the conditions c.a=9 and c.b=-4 . Then c is equal to

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Q. Given the vectors $\overrightarrow{a}=(3, -1,5)$ and $\overrightarrow{b}=(1,2,-3)$ . A vector $\overrightarrow{c}$ is such that it is perpendicular to the $z-axis$ and satisfies the conditions $\overrightarrow{c}.\overrightarrow{a}=9$ and $\overrightarrow{c}.\overrightarrow{b}=-4$ . Then $\overrightarrow{c}$ is equal to

Vector Algebra

Solution:

$\vec{c}$ can be $\left(- 2, 3, 0\right)$ or $\left(2, - 3, 0\right)$ since it is $\bot$ to $\left(0, 0, 1\right)$ i.e., $z$-axis
Since $\vec{c} . \vec{a}=9$ and $\vec{c} . \vec{b}=-4$
$\therefore \vec{c}=\left(2, -3, 0\right)$
$\bigg[\because (2,-3,0)\cdot(3,-1,5) = 6 + 3 = 9$$(2,-3,0)\cdot(1,2,-3) = 2 - 6 = -4\bigg]$