Name: Geometric Properties of the Cross Product
Uploaded: 2019-07-19
Duration: 10 min 22 s
Description: We will learn how to find the area of a triangle with the given vertices in a vector and find the angle between two vectors using the cross product

Question

Find a unit vector perpendicular to the vectors

4 \hat{i} - \hat{j} + 3 \hat{k}

and

2 \hat{i} + 2 \hat{j} - \hat{k}

Accepted Answer

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

Now the cross product of these vectors will be

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

$\Rightarrow \vec{a} \times \vec{b} = (1 - 6) \hat{i} - ((- 4) - 6) \hat{j} + (8 + 2) \hat{k}$

$\Rightarrow \vec{a} \times \vec{b} = - 5 \hat{i} + 10 \hat{j} + 10 \hat{k}$

And the corresponding magnitude will be

$|\vec{a} \times \vec{b}| = \sqrt{{(- 5)}^{2} + {(10)}^{2} + {(10)}^{2}}$

$\Rightarrow |\vec{a} \times \vec{b}| = \sqrt{25 + 100 + 100}$

$\Rightarrow |\vec{a} \times \vec{b}| = \sqrt{225}$

$\Rightarrow |\vec{a} \times \vec{b}| = 15$

Now we know that vector product $\vec{a} \times \vec{b}$ will be perpendicular to the plane of $\vec{a}$ and $\vec{b}$ .

So the unit vector perpendicular to the plane of $\vec{a}$ and $\vec{b}$ is given as

$= \pm \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$

Now substituting the corresponding value, we get

$= \pm \frac{- 5 \hat{i} + 10 \hat{j} + 10 \hat{k}}{15}$

Or, the unit vector can be written as

$= \pm (- \frac{5}{15} \hat{i} + \frac{10}{15} \hat{j} + \frac{10}{15} \hat{k})$

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

Find a unit vector perpendicular to the vectors $4 \hat{i} - \hat{j} + 3 \hat{k}$ and $2 \hat{i} + 2 \hat{j} - \hat{k}$

Important Questions on Vector or Cross Product of Two Vectors

Learn and Prepare for any exam you want !

Find a unit vector perpendicular to the vectors 4i^-j^+3k^ and 2i^+2j^-k^

Important Questions on Vector or Cross Product of Two Vectors

Learn and Prepare for any exam you want !

Find a unit vector perpendicular to the vectors $4 \hat{i} - \hat{j} + 3 \hat{k}$ and $2 \hat{i} + 2 \hat{j} - \hat{k}$