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

If

\vec{a} = 7 \hat{i} + 3 \hat{j} - 5 \hat{k}, \vec{b} = 2 \hat{i} + 5 \hat{j} - \hat{k}

and

\vec{c} = - \hat{i} + 2 \hat{j} + 4 \hat{k}

. Find

(\vec{a} - \vec{b}) \times (\vec{c} - \vec{b})

Accepted Answer

Given: $\vec{a} = 7 \hat{i} + 3 \hat{j} - 5 \hat{k}, \vec{b} = 2 \hat{i} + 5 \hat{j} - \hat{k}$ and $\vec{c} = - \hat{i} + 2 \hat{j} + 4 \hat{k}$ .

First, we will find: $(\vec{a} - \vec{b})$

$(\vec{a} - \vec{b}) = (7 \hat{i} + 3 \hat{j} - 5 \hat{k}) - (2 \hat{i} + 5 \hat{j} - \hat{k}),$

$\Rightarrow (\vec{a} - \vec{b}) = (5 \hat{i} - 2 \hat{j} - 4 \hat{k}) . . . (1)$

Now we will find: $(\vec{c} - \vec{b})$

$(\vec{c} - \vec{b}) = (- \hat{i} + 2 \hat{j} + 4 \hat{k}) - (2 \hat{i} + 5 \hat{j} - \hat{k}),$

$\Rightarrow (\vec{c} - \vec{b}) = (- 3 \hat{i} - 3 \hat{j} + 5 \hat{k}) . . . (2)$

Now we will find: $(\vec{a} - \vec{b}) \times (\vec{c} - \vec{b})$

We will substitute values from Equations $(1)$ and $(2)$ . Hence

$(\vec{a} - \vec{b}) \times (\vec{c} - \vec{b}) = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 5 & - 2 & - 4 \\ - 3 & - 3 & 5 \end{matrix}|,$

$\Rightarrow (\vec{a} - \vec{b}) \times (\vec{c} - \vec{b}) = ((- 10) - 12) \hat{i} - (25 - 12) \hat{j} + ((- 15) - 6) \hat{k},$

$\Rightarrow (\vec{a} - \vec{b}) \times (\vec{c} - \vec{b}) = - 22 \hat{i} - 13 \hat{j} - 21 \hat{k .}$

If $\vec{a} = 7 \hat{i} + 3 \hat{j} - 5 \hat{k}, \vec{b} = 2 \hat{i} + 5 \hat{j} - \hat{k}$ and $\vec{c} = - \hat{i} + 2 \hat{j} + 4 \hat{k}$ . Find $(\vec{a} - \vec{b}) \times (\vec{c} - \vec{b})$

Important Questions on Vector or Cross Product of Two Vectors

Learn and Prepare for any exam you want !

If a→=7i^+3j^-5k^, b→=2i^+5j^-k^ and c→=-i^+2j^+4k^. Find a→-b→×c→-b→

Important Questions on Vector or Cross Product of Two Vectors

Learn and Prepare for any exam you want !

If $\vec{a} = 7 \hat{i} + 3 \hat{j} - 5 \hat{k}, \vec{b} = 2 \hat{i} + 5 \hat{j} - \hat{k}$ and $\vec{c} = - \hat{i} + 2 \hat{j} + 4 \hat{k}$ . Find $(\vec{a} - \vec{b}) \times (\vec{c} - \vec{b})$