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51.
Let A be a point in space such that $|OA| = 12$ where O is the origin. If $OA$ is inclined at angles 45^o and 60⁰ with x-axis and y-axis respectively, then what is $OA$ equal to ?

$6 \hat{j} + 6 \hat{i} + 2 \hat{k}$

$6 \sqrt{2} \hat{i} + 6 \hat{j} \pm 6 \hat{k}$

$6 \sqrt{2} \hat{j} + 6 \hat{i} \pm 6 \hat{k}$

$3 \sqrt{2} \hat{j} + 3 \hat{i} \pm 6 \hat{k}$

$6 \sqrt{2} \hat{j} + 6 \hat{i} \pm 6 \hat{k}$

Let α, β and ү are the angles vector OA makes with the x, y and z axis respectively

Then,

$\cos^{2} (α) + \cos^{2} (β) + \cos^{2} (γ) = 1 \frac{1}{2} + \frac{1}{4} + \cos^{2} (γ) = 1 [α = 45 °, β = 60 °] \cos^{2} (γ) = \frac{1}{4} \cos (γ) = \pm \frac{1}{2} It is given that OA = |OA| (\cos (α) \hat{i} + \cos (β) \hat{j} + \cos (γ) \hat{k}) = 6 \sqrt{2} \hat{j} + 6 \hat{i} \pm 6 \hat{k}$