---
title: "Vector dot and cross product problem"
subject: "Computer Aided Design"
chapter: 2
key: "semester_4/computer_aided_design:vector-algebra-numerical"
source: https://www.bctengineer.com/q/semester_4/computer_aided_design/vector-algebra-numerical
site: BCT Engineering
---

# Vector dot and cross product problem

Computer Aided Design · Chapter 2: Geometric Modeling Fundamentals

- Asked in: year not given, 5 marks
- Video: [Dot Product and the Cross Product |Simulations | Mechanics | Mechanical Engineering | LUT University](https://www.youtube.com/watch?v=6nPVzWaPTJY) — TEMEKO
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/computer_aided_design/vector-algebra-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-2

## Question

Two vectors are A = 2i + 3j + k and B = 4i - j + 2k. Find (a) A.B, (b) A x B, (c) the angle between them, (d) the unit vector normal to the plane of A and B, and (e) the area of the parallelogram formed by A and B. State where each result is used in geometric modelling.

## Short answer

A.B = 7; A x B = (7, 0, -14); angle = 65.9 deg; unit normal = (0.447, 0, -0.894); parallelogram area = 15.65 square units. Dot product is used for angles and projections; cross product gives surface normals and areas.

## Answer

#### Given
$\mathbf A = (2, 3, 1)$, $\mathbf B = (4, -1, 2)$.

#### (a) Dot product
$$
\mathbf A\cdot\mathbf B = 2(4)+3(-1)+1(2) = 8-3+2 = 7
$$

#### (b) Cross product
$$
\mathbf A\times\mathbf B=
\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\2&3&1\\4&-1&2\end{vmatrix}
= (6+1)\mathbf i-(4-4)\mathbf j+(-2-12)\mathbf k = 7\mathbf i+0\mathbf j-14\mathbf k
$$

#### (c) Angle
$|\mathbf A|=\sqrt{14}=3.742$, $|\mathbf B|=\sqrt{21}=4.583$.

$$
\cos\theta=\frac{7}{3.742\times4.583}=0.4082,\qquad \theta=65.9^\circ
$$

#### (d) Unit normal
$|\mathbf A\times\mathbf B|=\sqrt{49+0+196}=15.652$

$$
\hat{\mathbf n}=\frac{(7,0,-14)}{15.652}=(0.447,\ 0,\ -0.894)
$$

#### (e) Area of parallelogram
Area $=|\mathbf A\times\mathbf B| = 15.65$ square units.

#### Uses in modelling
- Dot product: angle between edges, projection of a point on a line, checking perpendicularity (dot product = 0).
- Cross product: surface normal of a face (used in shading, offsetting, STL facet normals), area of a face, and the sign of a turn.

**Answer: A.B = 7; AxB = (7, 0, -14); angle = 65.9 deg; unit normal = (0.447, 0, -0.894); area = 15.65 square units.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is A.B for A = (2,3,1), B = (4,-1,2)?
   A. 13
   B. 9
   C. 7
   D. -7
   Answer: C. 2(4) + 3(-1) + 1(2) = 8 - 3 + 2 = 7. (Key point: Dot product)

2. What is A x B?
   A. (7, 0, 14)
   B. (-7, 0, -14)
   C. (7, 14, 0)
   D. (7, 0, -14)
   Answer: D. Determinant gives (6+1)i - (4-4)j + (-2-12)k = (7, 0, -14). (Key point: Cross product)

3. How is the parallelogram area found?
   A. |A x B| = 15.65
   B. A.B
   C. |A||B| = 17.15
   D. |A x B|/2
   Answer: A. The area of the parallelogram equals the magnitude of the cross product. (Key point: Area)

4. Which use of the cross product is given?
   A. Checking perpendicularity
   B. Surface normal of a face
   C. Finding the projection on a line
   D. Finding an edge length
   Answer: B. The cross product gives face normals (shading, offsetting, STL); the dot product tests perpendicularity. (Key point: Uses)

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