Question

Given A= 2i + 4j – 3k , B = 6i-4j+3k , C= 2i-2j+3k

Show that (i) A. B = B.A (ii) A*B = - B*A

(iii) A. (B+C) = A.B + A.C (iv) A* (B+C) = A*B + A*C

Answer 1

i)

$$\vec{A}\cdot\vec{B}=(2,4,-3)(6,-4,3)=-13\\ \vec{B}\cdot\vec{A}=(6,-4,3)(2,4,-3)=-13=\vec{A}\cdot\vec{B}$$

ii)

$$\vec{A}\times\vec{B}=(2,4,-3)\times(6,-4,3)=(0,-24,-32)\\ \vec{B}\times\vec{A}=(6,-4,3)\times(2,4,-3)=(0,24,32)\\=-\vec{A}\times\vec{B}$$

iii)

$$\vec{A}\cdot(\vec{B}+\vec{C})=(2,4,-3)(8,-6,6)=-26\\ \vec{A}\cdot\vec{C}=(2,4,-3)(2,-2,3)=-13\\\vec{A}\cdot\vec{B}=(2,4,-3)(6,-4,3)=-13\\-26=-13-13$$

iv)

$$\vec{A}\times\vec{B}=(2,4,-3)\times(6,-4,3)=(0,-24,-32)\\\vec{A}\times\vec{C}=(2,4,-3)\times(2,-2,3)=(6,-12,-12)\\\vec{A}\times(\vec{B}+\vec{C})=(2,4,-3)\times(8,-6,6)\\=(6,-36,-44) \\(0,-24,-32)+(6,-12,-12)=(6,-36,-44)$$