Vectors Problem Example 4

Description:

Consider following vectors −

a→ = 3î - 4ĵ + 5k̂

b→ = -2î + ĵ - 3k̂

1. Find the dot product, P = a→ . b→

1. Find the cross product, P = a→ × b→

P = a→ . b→

P = (3î - 4ĵ + 5k̂) . (-2î + ĵ - 3k̂)

Now, we know,

î . î = ĵ . ĵ = k̂ . k̂ = 1

î . ĵ = ĵ . k̂ = k̂ . î = ĵ . î = k̂ . ĵ = î . k̂ = 0

Hence,

P = -6 - 4 - 15

P = -25 .... Answer

P→ = a→ . b→

P→ = (3î - 4ĵ + 5k̂) × (-2î + ĵ - 3k̂)

Now, we know,

î × î = ĵ × ĵ = k̂ × k̂ = 0

î × ĵ = k̂ ; ĵ × k̂ = î ; k̂ × î = ĵ

ĵ × î = -k̂ ; k̂ × ĵ = -î ; î × k̂ = -ĵ

Diagram Explained

Hence,

P→ = 3k̂ + 9ĵ - 8k̂ + 12î - 10ĵ - 5î

P→ = 7î - ĵ - 5k̂ .... Answer

P→ = a→ . b→

P→ = (3î - 4ĵ + 5k̂) × (-2î + ĵ - 3k̂)

Creating the determinant,

Matrix

P→ = î [(-4)(-3) - (5)(1)] - ĵ [(3)(-3) - (-2)(5)] + k̂ [(3)(1) - (-2)(-4)]

P→ = 7î - ĵ - 5k̂ ... Answer

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