# Find All Vectors of Magnitude 10 √ 3 that Are Perpendicular to the Plane of ^ I + 2 ^ J + ^ K and − ^ I + 3 ^ J + 4 ^ K . - Mathematics

Sum

Find all vectors of magnitude $10\sqrt{3}$ that are perpendicular to the plane of $\hat{ i } + 2 \hat{ j } + \hat{ k }$ and $- \hat { i } + 3 \hat{ j } + 4 \hat{ k }$ .

#### Solution

Let $\vec{a} = \hat{ i } + 2 \hat{ j } + \hat{ k }$ and $\vec{b} = - \hat{ i } + 3 \hat{ j } + 4 \hat{ k }$ .

Unit vectors perpendicular to both $\vec{a}$ and  $\vec{b}$ =  $\pm \frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|}$

Now,

$\vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 1 & 2 & 1 \\ - 1 & 3 & 4\end{vmatrix} = 5 \hat{ i } - 5 \hat{ j } + \hat{ k }$

$\therefore \left| \vec{a} \times \vec{b} \right| = \left| 5 \hat{ i } - 5 \hat{ j } + 5 \hat{ k } \right| = \sqrt{5^2 + \left( - 5 \right)^2 + 5^2} = \sqrt{75} = 5\sqrt{3}$

Unit vectors perpendicular to both $\vec{a}$ and $\vec{b}$ =  $\pm \frac{5 \hat{ i } - 5 \hat{ j } + 5 \hat{ k } }{5\sqrt{3}} = \pm \frac{\hat{ i } - \hat{ j } + \hat{ k } }{\sqrt{3}}$

∴ Required vectors = $10\sqrt{3}\left( \pm \frac{\hat{ i} - \hat{ j } + \hat{ k } }{\sqrt{3}} \right) = \pm 10\left( \hat{ i } - \hat{ j } + \hat{ k } \right)$

Thus, the vectors of magnitude $10\sqrt{3}$  that are perpendicular to the plane of $\hat{ i } + 2 \hat{ j } + \hat{ k }$ and  $- \hat{ i } + 3 \hat{ j } + 4 \hat{ k }$ are  $\pm 10\left( \hat{ i } - \hat{ j } + \hat{ k } \right)$ .

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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 25 Vector or Cross Product
Exercise 25.1 | Q 35 | Page 31