Question

Use the geometric definition of the cross product and the properties of the cross product to make the following calculations. (a) ((i⃗ +j⃗ )×i⃗ )×j⃗ = (b) (j⃗ +k⃗ )×(j⃗ ×k⃗ ) = (c) 4i⃗ ×(i⃗ +j⃗ ) = (d) (k⃗ +j⃗ )×(k⃗ −j⃗ ) =

Answer 1

Answer:

Step-by-step explanation:

we know

$\vec{i}\times \vec{j}=\vec{k}$

$\vec{j}\times \vec{k}=\vec{i}$

$\vec{k}\times \vec{i}=\vec{j}$

(a)$\left [ \left ( \hat{i}+\hat{j}\right )\times \hat{i}\right ]\times \hat{j}$

$=\left [ \hat{i}\times \hat{i}+\hat{j}\times \hat{i}\right ]\times \hat{j}$

$=\left [ 0-\hat{k}\right ]\times \hat{j}$

$=\hat{i}$

(b)$\left ( \hat{j}+\hat{k}\right )\times \left ( \hat{j}\times \hat{k}\right )$

$=\left ( \hat{j}+\hat{k}\right )\times \left ( \hat{i}\right )$

$=\hat{k}+\hat{j}$

(c)$4\hat{i}\times \left ( \hat{i}+\hat{j}\right )$

$=4\hat{i}\times \hat{i}+4\hat{i}\times \hat{j}$

$=0+4\hat{k}$

(d)$\left ( \hat{k}+\hat{j}\right )\times \left ( \hat{k}-\hat{j}\right )$

$=\hat{k}\times \hat{k}-\hat{k}\times \hat{j}+\hat{j}\times \hat{k}-\hat{j}\times \hat{j}$

$=0+\hat{i}+\hat{i}-0$

$=2\hat{i}$