Vector triple product involves three vectors and two cross products. It is obtained by taking the cross product of one vector with the cross product of two other vectors. The result is another vector that can be simplified using the vector triple product identity.
For vectors,
a × (b × c) = (a ⋅ c)b - (a ⋅ b)c
Problem 1: Given three vectors a = 2i - 3j + 4k, b = -i + 2j - 3k, and c = i + 3j - 2k, find (a × b) × c.
First, let's find a × b:
a × b = (2i - 3j + 4k) × (-i + 2j - 3k)
= i[(-3)(-3) - 4(2)] - j[2(-3) - 4(-1)] + k[2(2) - (-3)(-1)]
= i(9 - 8) - j(-6 + 4) + k(4 - 3)
= i(1) + j(2) + k(1)
= i + 2j + k
Now, compute (a × b) × c:
(a × b) × c = (i + 2j + k) × (i + 3j - 2k)
= i[2(-2) - 1(3)] - j[1(-2) - 1(1)] + k[1(3) - 2(1)]
= i(-4 - 3) - j(-2 - 1) + k(3 - 2)
= -7i + 3j + k
So, (a × b) × c = -7i + 3j + k
Problem 2: Given vectors p = 3i + 2j - k, q = 2i - 4j + 3k, and r = -i + 3j + 2k, find (p × q) × r.
First, let's find p × q:
p × q = (3i + 2j - k) × (2i - 4j + 3k)
= i[(2)(3) - (-1)(-4)] - j[(3)(3) - (-1)(2)] + k[(3)(-4) - (2)(2)]
= i(6 - 4) - j(9 + 2) + k(-12 - 4)
= 2i - 11j - 16k
Now, compute (p × q) × r:
(p × q) × r = (2i - 11j - 16k) × (-i + 3j + 2k)
= i[(-11)(2) - (-16)(3)] - j[(2)(2) - (-16)(-1)] + k[(2)(3) - (-11)(-1)]
= i(-22 + 48) - j(4 - 16) + k(6 - 11)
= 26i + 12j - 5k
So, (p × q) × r = 26i + 12j - 5k.
Problem 3: Given vectors u = 5i - 2j + 3k, v = 4i - j + 2k, and w = 2i + j - k, find (u × v) × w.
First, let's compute u × v:
u × v = (5i - 2j + 3k) × (4i - j + 2k)
= i[(-2)(2) - (3)(-1)] - j[(5)(2) - (3)(4)] + k[(5)(-1) - (-2)(4)]
= i(-4 + 3) - j(10 - 12) + k(-5 + 8)
= -i + 2j + 3k
Now, compute (u × v) × w:
(u × v) × w = (-i + 2j + 3k) × (2i + j - k)
= i[(2)(-1) - (3)(1)] - j[(-1)(-1) - (3)(2)] + k[(-1)(1) - (2)(2)]
= i(-2 - 3) - j(1 - 6) + k(-1 - 4)
= -5i + 5j - 5k
So, (u × v) × w = -5i + 5j - 5k.
Problem 4: Given vectors x = i - 3j + 2k, y = -2i + j + 3k, and z = 4i + j - k, find (x × y) × z.
First, let's compute x × y:
x × y = (i - 3j + 2k) × (-2i + j + 3k)
= i[(-3)(3) - (2)(1)] - j[(1)(3) - (2)(-2)] + k[(1)(1) - (-3)(-2)]
= i(-9 - 2) - j(3 + 4) + k(1 - 6)
= -11i - 7j - 5k
Now, compute (x × y) × z:
(x × y) × z = (-11i - 7j - 5k) × (4i + j - k)
= i[(-7)(-1) - (-5)(1)] - j[(-11)(-1) - (-5)(4)] + k[(-11)(1) - (-7)(4)]
= i(7 + 5) - j(11 + 20) + k(-11 + 28)
= 12i - 31j + 17k
So, (x × y) × z = 12i - 31j + 17k.
Problem 5: Given vectors m = i + 2j - 3k, n = 3i - j + 2k, and o = -i + 3j + 2k, find (m × n) × o.
First, let's compute m × n:
m × n = (i + 2j - 3k) × (3i - j + 2k)
= i[(2)(2) - (-3)(-1)] - j[(1)(2) - (-3)(3)] + k[(1)(-1) - (2)(3)]
= i(4 - 3) - j(2 + 9) + k(-1 - 6)
= i - 11j - 7k
Now, compute (m × n) × o:
(m × n) × o = (i - 11j - 7k) × (-i + 3j + 2k)
= i[(-11)(2) - (-7)(3)] - j[(1)(2) - (-7)(-1)] + k[(1)(3) - (-11)(-1)]
= i(-22 + 21) - j(2 - 7) + k(3 - 11)
= -i + 5j - 8k
So, (m × n) × o = -i + 5j - 8k.
Problem 6: Given vectors a = i + 2j + 3k, b = 4i + 5j + 6k, and c = 7i + 8j + 9k, find (a × b) × c.
First, compute a × b:
a × b = (i + 2j + 3k) × (4i + 5j + 6k)
= i[(2)(6) - (3)(5)] - j[(1)(6) - (3)(4)] + k[(1)(5) - (2)(4)]
= i(12 - 15) - j(6 - 12) + k(5 - 8)
= -3i + 6j - 3k
Now, compute (a × b) × c:
(a × b) × c = (-3i + 6j - 3k) × (7i + 8j + 9k)
= i[(6)(9) - (-3)(8)] - j[(-3)(9) - (-3)(7)] + k[(-3)(8) - (6)(7)]
= i(54 + 24) - j(-27 + 21) + k(-24 - 42)
= 78i + 6j - 66k
So, (a × b) × c = 78i + 6j - 66k.
Vector Triple Product Worksheet
Q1. Given vectors a = i + j + k, b = 7i - j + 6k, and c = i + 8j - 9k, find (a × b) × c.
Q2. Given vectors a = 2i + 2j + 2k, b = 4i + 5j + 6k, and c = i - 8j + 9k, find (a × b) × c.
Q3. Prove the Vector Triple Product Identity
Q4. Prove that if a, b, and c are mutually orthogonal unit vectors, then a × (b × c) = -c
Q5. Given vectors a = i - j + k, b = i + j + k, and c = 7i + 8j - 9k, find (a × b) × c.
