Let a unit vector which makes an angle of 60º with and an angle of 45º with . Then is :

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Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

Let a unit vector which makes an angle of 60º with $2 \hat{i} + 2 \hat{j} - \hat{k}$ and an angle of 45º with $\hat{i} - \hat{k} be \vec{C}$ . Then $\vec{C} + (- \frac{1}{2} \hat{i} + \frac{1}{3 \sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k})$ is :

$\frac{\sqrt{2}}{3} \hat{i} - \frac{1}{2} \hat{k}$

$- \frac{\sqrt{2}}{3} \hat{i} + \frac{\sqrt{2}}{3} \hat{j} + (\frac{1}{2} + \frac{2 \sqrt{2}}{3}) \hat{k}$

$\frac{\sqrt{2}}{3} \hat{i} + \frac{1}{3 \sqrt{2}} \hat{j} - \frac{1}{2} \hat{k}$

$(\frac{1}{\sqrt{3}} + \frac{1}{2}) \hat{i} + (\frac{1}{\sqrt{3}} - \frac{1}{3 \sqrt{2}}) \hat{j} + (\frac{1}{\sqrt{3}} + \frac{\sqrt{2}}{3}) \hat{k}$

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Let $\hat{C} = x \hat{i} + y \hat{j} + z \hat{k}$

$| \hat{C} | = 1 \Rightarrow x^{2} + y^{2} + z^{2} = 1$ ---------------- (1)

as angle between $\hat{C}$ and $2 \hat{i} + 2 \hat{j} - \hat{k} is 60^{\circ}$

$\Rightarrow \cos 60^{\circ} = \frac{2 x + 2 y - z}{3}$

⇒ 4x + 4y – 2z = 3 -------- (2)

and angle between $\hat{C} and (\hat{i} - \hat{k}) is 45^{\circ}$

$\Rightarrow \cos 45^{\circ} = \frac{x - z}{\sqrt{2}}$

⇒ x – z = 1 ------- (3)

by (2) and (3)

4y + 2z + 1 = 0

$y = - (\frac{2 z + 1}{4})$

by $(1) (z + 1)^{2} + {(\frac{2 z + 1}{- 4})}^{2} + z^{2} = 1$

⇒ 16 (z² + 2z + 1) + (4z² + 4z +1) + 16z² = 16
⇒ 36z² + 36z + 1 = 0

$\Rightarrow z = \frac{- 36 \pm \sqrt{1296 - 144}}{72}$

$\Rightarrow z = \frac{- 36 \pm \sqrt{1152}}{72}$

$\Rightarrow z = - \frac{1}{2} \pm \frac{24 \sqrt{2}}{72}$

$\Rightarrow z = - \frac{1}{2} \pm \frac{\sqrt{2}}{3}$

$\Rightarrow \hat{C} = (z + 1) \hat{i} - \frac{1}{4} (2 z + 1) \hat{j} + z \hat{k}$

$\Rightarrow \vec{C} + (- \frac{1}{2} \hat{i} + \frac{1}{3 \sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k}) = (z + \frac{1}{R}) \hat{i} - \frac{1}{2} (z + \frac{1}{2}) \hat{j} + z \hat{k} + \frac{1}{3 \sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k}$

$= \frac{\sqrt{2}}{3} \hat{i} - \frac{1}{3 \sqrt{2}} \hat{j} + (- \frac{1}{2} + \frac{\sqrt{2}}{3}) \hat{k} + \frac{1}{3 \sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k}$ $= \frac{\sqrt{2}}{3} \hat{i} - \frac{1}{2} \hat{k} is$

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