Vector triple product involves three vectors. When you take the cross product of the vectors B and C, and then take the cross product of the result with vector A, it gives a vector perpendicular to both B and C, with its magnitude dependent on A and the angle between B and C.
Vector Triple Product Formula
For vectors,
a × (b × c) = (a ⋅ c)b - (a ⋅ b)c
Practice Problems on Vector Triple Product
Problem 1: Given three vectors a = 2i - 3j + 4k, b = -i + 2j - 3k, and c = i + 3j - 2k, find (a × b) × c.
Solution:
First, let's find a × b:
a × b = (2i - 3j + 4k) × (-i + 2j - 3k)
= (j × k)(2(-3) - 4(2)) - (i × k)(-2 - (-1)(-3)) + (i × j)(-2 - (-3)(-3))
= i(-3) - j(8) + k(1)
= -3i - 8j + k
Now, let's compute (a × b) × c:
(a × b) × c = (-3i - 8j + k) × (i + 3j - 2k)
= (j × k)(-2(-8) - 1(3)) - (i × k)(1 - (-3)(-3)) + (i × j)(1(3) - (-2)(-8))
= i(-23) - j(4) + k(7)
= -23i - 4j + 7k
So, (a × b) × c = -23i - 4j + 7k
Problem 2: Given vectors p = 3i + 2j - k, q = 2i - 4j + 3k, and r = -i + 3j + 2k, find (p × q) × r.
Solution:
First, let's find p × q: p × q = (2i - 13j + 10k)
Now, let's compute (p × q) × r: (p × q) × r = (-23i - 4j + 7k)
So, (p × q) × r = -23i - 4j + 7k
Problem 3: Given vectors u = 5i - 2j + 3k, v = 4i - j + 2k, and w = 2i + j - k, find (u × v) × w.
Solution:
Let's first compute u × v: u × v = (i + 2j + 17k)
Next, find (u × v) × w: (u × v) × w = (35i - 5j + 5k)
So, (u × v) × w = 35i - 5j + 5k
Problem 4: Given vectors x = i - 3j + 2k, y = -2i + j + 3k, and z = 4i + j - k, find (x × y) × z.
Solution:
Compute x × y: x × y = (7i - j + 5k)
Now, find (x × y) × z: (x × y) × z = (-9i - 31j + 3k)
So, (x × y) × z = -9i - 31j + 3k
Problem 5: Given vectors m = i + 2j - 3k, n = 3i - j + 2k, and o = -i + 3j + 2k, find (m × n) × o.
Solution:
Compute m × n: m × n = (5i - 11j + 5k)
Now, find (m × n) × o: (m × n) × o = (-9i - 31j + 3k)
So, (m × n) × o = -9i - 31j + 3k
Problem 6: Given vectors a = i + 2j + 3k, b = 4i + 5j + 6k, and c = 7i + 8j + 9k, find (a × b) × c.
Solution:
Compute b × c:
b × c =
\begin{bmatrix} i & j& k\\ 4 & 5& 6\\ 7 & 8& 9 \end{bmatrix} = i(5⋅9 - 6⋅8) - j(4⋅9 - 6⋅7) + k(4⋅8 - 5⋅7)
= i(45 - 48) - j(36 - 42) + k(32 - 35)
= i(-3) - j(-6) + k(-3)
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.
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Conclusion
Vector triple product, a × (b × c), simplifies to (a ⋅ c)b - (a ⋅ b)c, yielding a vector perpendicular to both b and c. This formula helps solve problems involving three vectors. Practice problems illustrate its application, confirming that careful step-by-step calculations lead to accurate results. Understanding this product is essential in vector algebra.