Question

$Physics \ question\ vectors \ 3d$
Find the cross product and dot product of the given vectors and check if the answer you got is different or same.

A = 3i + 2j + 6k and B = 5i + 7j + 3k

Mordetor please solve

Answer 1

Answer:

Cross product = -36i + 21j + 11k

Dot product = 47 units .

Step-by-step explanation

Cross product :

For cross product refer to the attachment.

Cross product is -36i + 21j + 11k .

Dot product :

We have,

A = 3i + 2j + 6k

B = 5i + 7j + 3k

We know,

→ A.B = Ax Bx + Ay By + Az Bz

→ A.B = (3 × 4) + (2 × 7) + (6 × 3)

→ A.B = 15 + 14 + 18

→ A.B = 47

Dot product is 47 units.

The answer of cross product and dot product are different.

Answer 2

EXPLANATION.

Cross products and dot products.

$\sf \implies A = 3\hat{i} + 2 \hat{j} + 6 \hat{k}$

$\sf \implies B = 5\hat{i} + 7 \hat{j} + 3 \hat{k}$

As we know that,

For scalar products.

$\sf If \ \vec{a} = a_{1}\hat{i} + a_{2}\hat{j} + a_{3} \hat{k} \ \ and \ \ \vec{b} = b_{1}\hat{i} + b_{2}\hat{j} + b_{3}\hat{k}$

$\sf \implies \vec{a} \times \vec{b} = \left[\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{array}\right]$

Using this formula in the equation, we get.

⇒ a₁ = 3, a₂ = 2, a₃ = 6.

⇒ b₁ = 5, b₂ = 7, b₃ = 3.

Put the values in the equation, we get.

$\sf \implies \vec{a} \times \vec{b} = \left[\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\3&2&6\\5&7&3\end{array}\right]$

$\sf \implies \vec{a} \times \vec{b} =\hat{i} [2(3) - 6(7)] - \hat{j} [3(3) - 6(5)] + \hat{k} [3(7) - 2(5)]$

$\sf \implies \vec{a} \times \vec{b} = \hat{i}[6 - 42] - \hat{j} [9 - 30 ] + \hat{k} [21 - 10]$

$\sf \implies \vec{a} \times \vec{b} = - 36 \hat{i} + 21 \hat{j} + 11 \hat{k}$

For dot products.

$\sf \implies A = 3\hat{i} + 2 \hat{j} + 6 \hat{k}$

$\sf \implies B = 5\hat{i} + 7 \hat{j} + 3 \hat{k}$

$\sf \implies A.B = (3 \hat{i} + 2 \hat{j} + 6 \hat{k} )(5 \hat{i} + 7 \hat{j} + 3 \hat{k} )$

$\sf \implies A.B = [(3 \times 5) + (2 \times 7) + (6 \times 3)].$

$\sf \implies A.B = [15 + 14 + 18].$

$\sf \implies A.B = 47.$

                                                                                                                   

MORE INFORMATION.

(1) Maximum values of   $\vec{a} . \vec{b} = | \vec{a} | | \vec{b}|$

(2) Minimum values of   $\vec{a} . \vec{b} = - | \vec{a} | | \vec{b} |$

(3) Any vector $\vec{a}$ can be written as,   $\vec{a} = (\vec{a} . \hat{i} )\hat{i} + (\vec{a} . \hat{j} )\hat{j} + ( \vec{a} . \hat{k} )\hat{k}$

(4) A vector in the direction of the bisector of the angle between the two vectors $\vec{a} \ and \ \vec{b} \ is \ \frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|}$  Hence bisector of the angle between the two vectors $\vec{a} \ and \ \vec{b} \ is \ \lambda ( \hat{a} + \hat{b} )$  where λ ∈ R. Bisector of the exterior angle between $\vec{a} \ and \ \vec{b} \ is \ \lambda (\hat{a} - \hat{b})$  λ ∈ R.