Question

Two vectors are given by a 3 i 4 j and b 2 i 5 j calculate their dot and cross products.

Answer 1

Dot product = - 14 and cross product = - 23k

Step-by-step explanation:

Given,

Vector a = 3i + 4j and vector b = 2i - 5j

To find, the dot  and cross products = ?

∴ Dot product = A.B

= (3i)(2i) + (4j)(- 5j)

= 6 - 20 [ ∵ i·i = 1]

= - 14

Also,

Cross product = A × B

$\left[\begin{array}{ccc}i&j&k\\3&4&0\\2&-5&0\end{array}\right]$

= i(0 + 0) - j(0 + 0) + k(- 15 -8)

= 0 + 0 + k( - 23)

= - 23k

∴ Dot product = - 14 and cross product = - 23k

Answer 2

$-23\hat{k}$

Step-by-step explanation:

Given,

$\vec{a} = 3\hat{i}+4\hat{j}$

$\vec{b}= 2\hat{i}-5\hat{j}$

dot product

$\vec{a}\cdot \vec{b}=( 3\vec{i}+4\vec{j}) (2\vec{i}-5\vec{j})$

$6\vec{i}^2-20\vec{j}^2$                    

$6\times1-20\times1$            $\therefore \vec{i}\cdot\vec{i}=1$

                               $\therefore\vec{j}\cdot\vec{j}=1$

cross product

$\vec{a}\times\vec{b}= \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\3 & 4 & 0 &\\2 & -5 & 0\end{vmatrix}$

       $= (-5\times 3)\hat{i} -( 4\times 2)\hat{j}$

       = $-23\hat{k}$      $\because \hat{i}\times\hat{j} = \hat{k}$