1st Year Math Chapter 14 Exercise 14.4 Notes: Scalar Triple Product and Applications (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 14.4 closes the chapter and the textbook by combining the dot product and cross product from Exercises 14.2 and 14.3 into the scalar triple product, used to find volumes and test whether three vectors lie in the same plane.
What Does This Exercise Cover?
This exercise teaches how to compute the scalar triple product of three vectors, its geometric meaning as a volume, and how it’s used to determine whether three given vectors are coplanar.
Key Concepts
1. The Scalar Triple Product
For three vectors u, v, and w, the scalar triple product is defined as u·(v × w), often written [u v w]. It combines a cross product (giving a vector) with a dot product (giving a final scalar result).
2. Determinant Formula
If u, v, and w are written in component form, the scalar triple product can be computed directly as the determinant of the 3×3 matrix formed by stacking their components as rows.
3. Volume of a Parallelepiped
If u, v, and w represent three edges of a parallelepiped meeting at a common vertex, the volume of that parallelepiped is Volume = |u·(v × w)|.
4. Coplanar Vectors
Three vectors are coplanar (all lying in the same plane) if and only if their scalar triple product equals zero — geometrically, this means the ‘parallelepiped’ they form has been flattened to zero volume.
5. Volume of a Tetrahedron
A tetrahedron formed by the same three edge vectors has one-sixth the volume of the corresponding parallelepiped: Volume = (1/6)|u·(v × w)|.
Solved Examples
Example 1: Find the scalar triple product of u = (1, 0, 0), v = (0, 1, 0), and w = (0, 0, 1).
First find v × w: using the determinant formula, v × w = (1, 0, 0).
Then u·(v × w) = (1,0,0)·(1,0,0) = 1(1) + 0(0) + 0(0).
Answer: 1
Example 2: Find the volume of the parallelepiped with edges u = (2, 0, 0), v = (0, 3, 0), and w = (0, 0, 4).
The scalar triple product is the determinant of the matrix with rows (2,0,0), (0,3,0), (0,0,4).
Since this is a diagonal matrix, the determinant is the product of the diagonal entries: 2 × 3 × 4.
Answer: Volume = 24 cubic units
Example 3: Determine whether u = (1, 2, 3), v = (2, 3, 4), and w = (3, 5, 7) are coplanar.
Compute the determinant of the matrix with rows (1,2,3), (2,3,4), (3,5,7).
= 1(3×7 − 4×5) − 2(2×7 − 4×3) + 3(2×5 − 3×3)
= 1(21−20) − 2(14−12) + 3(10−9) = 1 − 4 + 3.
Answer: The scalar triple product is 0, so the vectors are coplanar
Sample MCQs
1. The scalar triple product of vectors u, v, w is defined as:
a) u × (v × w) b) u·(v × w) c) u + v + w d) u·v·w
Answer: b) u·(v × w)
2. The scalar triple product gives:
a) A vector b) A scalar c) A matrix d) An angle
Answer: b) A scalar
3. The volume of a parallelepiped with edge vectors u, v, w is:
a) u·(v × w) b) |u·(v × w)| c) u × (v × w) d) |u| + |v| + |w|
Answer: b) |u·(v × w)|
4. Three vectors are coplanar if their scalar triple product equals:
a) 1 b) 0 c) −1 d) Their magnitudes
Answer: b) 0
5. The volume of a tetrahedron with edge vectors u, v, w from a common vertex is:
a) |u·(v × w)| b) (1/6)|u·(v × w)| c) (1/2)|u·(v × w)| d) 6|u·(v × w)|
Answer: b) (1/6)|u·(v × w)|
Important Short Questions
- Define the scalar triple product of three vectors.
- State the condition for three vectors to be coplanar, in terms of the scalar triple product.
- State the formula for the volume of a parallelepiped using the scalar triple product.
- State the formula for the volume of a tetrahedron using the scalar triple product.
- Explain why the scalar triple product results in a scalar, not a vector.
Important Long Questions
- Find the scalar triple product of u = (1, 2, 3), v = (0, 1, 4), and w = (2, 0, 1), using the determinant formula.
- Find the volume of the parallelepiped with edge vectors u = (2, 0, 0), v = (0, 3, 0), and w = (0, 0, 4).
- Determine whether u = (1, 2, 3), v = (2, 3, 4), and w = (3, 5, 7) are coplanar, using the scalar triple product.
- Explain the geometric meaning of the scalar triple product, connecting it to the volume of a parallelepiped.
How to Approach This Exercise Effectively
- Set up the scalar triple product directly as a 3×3 determinant rather than computing the cross product and dot product as two separate steps — it’s faster and less error-prone.
- Whenever a scalar triple product comes out to zero, immediately note the three vectors are coplanar — this is a common exam conclusion to state explicitly.
- Remember the tetrahedron volume is exactly one-sixth of the parallelepiped volume built from the same three edge vectors — don’t forget this factor.
- Take the absolute value at the very end when computing a volume — a negative determinant doesn’t mean a negative volume, just a different vector orientation.
- Practice expanding a 3×3 determinant by the same method every time (expansion along the first row) to build speed and consistency.
FAQs
Q: Why does a zero scalar triple product indicate coplanar vectors?
A: The scalar triple product measures the volume of the parallelepiped formed by three vectors; if that volume is zero, the vectors must be lying flat in a single plane rather than spanning three-dimensional space.
Q: What is the difference between the scalar triple product and the cross product?
A: The cross product combines two vectors into a new vector; the scalar triple product goes one step further, dotting that resulting vector with a third vector to produce a single number.
Q: How is the scalar triple product calculated using a determinant?
A: Writing the three vectors’ components as the three rows of a 3×3 matrix, the determinant of that matrix gives the scalar triple product directly, without needing to compute the cross product separately first.
Q: What real-world applications use this volume interpretation?
A: It’s used in physics and engineering to compute volumes of solids defined by edge vectors, and to test whether three force or direction vectors act within a single plane.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 14: Vectors in Space, Exercise 14.4.