Chapter 10: Vector Algebra
Key Formulas:
• |a + b|² = |a|² + |b|² + 2a·b
• a · b = |a||b| cos θ
• a × b = |a||b| sin θ n
• [a b c] = a · (b × c)
Q1. Find the unit vector in the direction of a = 2⋂⃗i + 3⋂⃗j − ⋂⃗k.
Solution
|a| = √(4 + 9 + 1) = √14
Unit vector = a/|a| = (2/√14)⋂⃗i + (3/√14)⋂⃗j − (1/√14)⋂⃗k
Q2. Find the dot product of a = ⋂⃗i + 2⋂⃗j − ⋂⃗k and b = 2⋂⃗i − ⋂⃗j + 3⋂⃗k.
Solution
a · b = (1)(2) + (2)(-1) + (-1)(3)
= 2 − 2 − 3 = −3
Q3. Find the angle between a = ⋂⃗i + ⋂⃗j − ⋂⃗k and b = ⋂⃗i − ⋂⃗j + ⋂⃗k.
Solution
a · b = 1 − 1 − 1 = −1
|a| = √3, |b| = √3
cos θ = (a · b)/(|a||b|) = −1/3
θ = cos−1(−1/3)
Q4. Find the cross product of a = 2⋂⃗i + 3⋂⃗j + ⋂⃗k and b = ⋂⃗i − ⋂⃗j + ⋂⃗k.
Solution
a × b = |⋂⃗i ⋂⃗j ⋂⃗k; 2 3 1; 1 −1 1|
= ⋂⃗i(3+1) − ⋂⃗j(2−1) + ⋂⃗k(−2−3)
= 4⋂⃗i − ⋂⃗j − 5⋂⃗k
Q5. Find the area of the parallelogram with adjacent sides a = 2⋂⃗i + 3⋂⃗j and b = ⋂⃗i + ⋂⃗j.
Solution
a × b = |⋂⃗i ⋂⃗j ⋂⃗k; 2 3 0; 1 1 0|
= ⋂⃗k(2 − 3) = −⋂⃗k
Area = |a × b| = |−⋂⃗k| = 1 sq. unit
Q6. Prove that [a + b, b + c, c + a] = 2[a, b, c].
Solution
[a+b, b+c, c+a] = (a+b) · [(b+c) × (c+a)]
(b+c) × (c+a) = b×c + b×a + c×c + c×a
= b×c − a×b + 0 + c×a
(a+b) · (b×c − a×b + c×a)
= a·(b×c) − a·(a×b) + a·(c×a) + b·(b×c) − b·(a×b) + b·(c×a)
Terms with two equal vectors = 0:
= [a,b,c] − 0 + 0 + 0 − 0 + [b,c,a]
= [a,b,c] + [a,b,c] = 2[a,b,c] ✓