Question

a , b and c are three vectors such that c is perpendicular to both a and b What is the value of a cross b cross c?

Answer 1

a × b × c = 0

Step-by-step explanation:

Given that,

• c and a are perpendicular to each other,

then c . a = 0

• c and b are perpendicular to each other,

then c . b = 0

Now, a × b × c

= (a × b) × c

= - c × (a × b)

= - {(c . b)a - (c . a)b}

= - {(0)a - (0)b}

= 0

Rules:

• a × (b × c) = (a . c)b - (a . b)c

• b × (c × a) = (b . a)c - (b . c)a

• c × (a × b) = (c . b)a - (c . a)b

• a . b = b . a, since (.) is commutative

• a × b = - b × a, since (×) is not commutative

Homework:

Let a = - 3i + 7j + 5k, b = - 5i + 7j - 3k, c = 7i - 5j - 3k. Find a × (b × c) in terms of i, j, k and then verify a × (b × c) = (a . c)b - (a . b)c.