Applications of vectors

Q: What is the condition for two non-zero vectors to be orthogonal (perpendicular)?

Two non-zero vectors \(\vec{a}\) and \(\vec{b}\) are orthogonal if and only if \(\vec{a} \cdot \vec{b} = 0\).

Q: How can you find the angle \(\theta\) between two non-zero vectors \(\vec{a}\) and \(\vec{b}\)?

\( \theta = \arccos\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right) \)

Q: What are the formulas for the scalar projection and vector projection of \(\vec{a}\) onto \(\vec{b}\)?

Scalar projection: \( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \); Vector projection: \( \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b} \)

Q: How is the cross product used to find the area of a parallelogram and a triangle formed by vectors \(\vec{a}\) and \(\vec{b}\)?

\( \text{Area of Parallelogram} = |\vec{a} \times \vec{b}| \) and \( \text{Area of Triangle} = \frac{1}{2}|\vec{a} \times \vec{b}| \)

Q: How is the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c})\) calculated using coordinates?

\( \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \)

Q: What are the vector formulas for the volume of a parallelepiped and a tetrahedron with coterminous edges \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\)?

\( V_{\text{parallelepiped}} = |\vec{a} \cdot (\vec{b} \times \vec{c})| \) and \( V_{\text{tetrahedron}} = \frac{1}{6}|\vec{a} \cdot (\vec{b} \times \vec{c})| \)

Q: What is the scalar triple product condition for three non-zero vectors to be coplanar?

\( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \)

Mathematics M2 (Algebra and Calculus) · Senior Secondary (HKDSE) · 10 min read · Updated: 24 Aug 2026

Applications of Vectors in M2

Hey everyone! Welcome to the study notes for "Applications of Vectors". You've learned what vectors are and how to do basic operations like adding and multiplying them. Now, let's unlock their true power! In this chapter, we'll see how vectors are not just abstract arrows but powerful tools for solving real geometric problems in 2D and 3D space. We'll learn how to find angles, project shadows, calculate areas and volumes, test coplanarity, and much more.

Don't worry if this sounds complicated. We'll break everything down into simple, step-by-step pieces. Think of it like adding new tools to your maths toolkit. Let's get started!

1. Division of a Line Segment

Imagine you have a line segment from point \(A\) to point \(B\). What if you want to find the exact coordinates of a point \(P\) along the line? This is where the section formula comes in, and it's super useful!

Internal Division

This is the most common case. We want to find a point \(P\) that lies between \(A\) and \(B\) and divides the line segment \(AB\) internally in the ratio \(m : n\).

Let \(\vec{a}\) and \(\vec{b}\) be the position vectors of points \(A\) and \(B\). The position vector of point \(P\), which divides \(AB\) internally in the ratio \(m : n\), is given by:

Section Formula (Internal Division):

\( \vec{p} = \frac{n\vec{a} + m\vec{b}}{m+n} \)

Memory Aid: The "Cross-Multiply" Trick

To remember the formula, think of it like this: the ratio part \(n\) multiplies with the vector \(\vec{a}\) (the one further away), and the ratio part \(m\) multiplies with the vector \(\vec{b}\) (the other one further away). Then, you divide by the sum of the ratios (\(m+n\)).

Example: Let \(A = (1, 2, 3)\) and \(B = (5, 6, 7)\). Find the point \(P\) that divides \(AB\) internally in the ratio \(1 : 3\).

Step 1: Identify your vectors and ratio.
\(\vec{a} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\), \(\vec{b} = \begin{pmatrix} 5 \\ 6 \\ 7 \end{pmatrix}\)
\(m = 1\), \(n = 3\)

Step 2: Apply the formula.
\( \vec{p} = \frac{3\vec{a} + 1\vec{b}}{1+3} = \frac{3\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + 1\begin{pmatrix} 5 \\ 6 \\ 7 \end{pmatrix}}{4} \)

Step 3: Calculate the result.
\( \vec{p} = \frac{\begin{pmatrix} 3 \\ 6 \\ 9 \end{pmatrix} + \begin{pmatrix} 5 \\ 6 \\ 7 \end{pmatrix}}{4} = \frac{\begin{pmatrix} 8 \\ 12 \\ 16 \end{pmatrix}}{4} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} \)
So, the coordinates of point \(P\) are \((2, 3, 4)\).

A Special Case: The Midpoint

The midpoint is just a point that divides the line segment in a \(1 : 1\) ratio. If we put \(m=1\) and \(n=1\) into the formula:

\( \vec{p} = \frac{1\vec{a} + 1\vec{b}}{1+1} = \frac{\vec{a} + \vec{b}}{2} \)

This is simply the average of the two position vectors!

External Division

When a point \(P\) lies on the line passing through \(A\) and \(B\) but outside the segment \(AB\) such that \(AP : PB = m : n\), it divides \(AB\) externally. The formula is:

Section Formula (External Division):

\( \vec{p} = \frac{-n\vec{a} + m\vec{b}}{m-n} \quad (m \neq n) \)

Key Takeaway

To find a point \(P\) dividing line segment \(AB\) internally in the ratio \(m:n\), use the formula \( \vec{p} = \frac{n\vec{a} + m\vec{b}}{m+n} \). For external division, replace \(n\) with \(-n\).

2. Parallelism and Orthogonality (Perpendicularity)

Vectors make it very easy to check the relationship between lines or directions. Are they parallel? Are they at a right angle? Let's see how.

Parallel Vectors

Two vectors are parallel if they point in the same direction or in exact opposite directions. Think of two parallel streets on a map.

The Rule: Two non-zero vectors \(\vec{a}\) and \(\vec{b}\) are parallel if one is a scalar multiple of the other.

\( \vec{a} = k \vec{b} \)

Here, \(k\) is just a non-zero real number (a scalar).

If \(k > 0\), they are in the same direction.
If \(k < 0\), they are in the opposite direction.

Example: Are the vectors \(\vec{a} = 2\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}\) and \(\vec{b} = -3\mathbf{i} - 6\mathbf{j} + 9\mathbf{k}\) parallel?

Let's see if we can find a \(k\) such that \(\vec{a} = k\vec{b}\).
Comparing the \(\mathbf{i}\) components: \(2 = k(-3) \implies k = -2/3\)
Comparing the \(\mathbf{j}\) components: \(4 = k(-6) \implies k = -2/3\)
Comparing the \(\mathbf{k}\) components: \(-6 = k(9) \implies k = -2/3\)
Since we get the same value of \(k\) for all components, the vectors are parallel! Because \(k\) is negative, they point in opposite directions.

Orthogonal (Perpendicular) Vectors

Two vectors are orthogonal (perpendicular) if the angle between them is \(90^\circ\).

The Rule: Two non-zero vectors \(\vec{a}\) and \(\vec{b}\) are orthogonal if and only if their dot product is zero.

\( \vec{a} \cdot \vec{b} = 0 \)

Why does this work?

Remember the dot product formula: \(\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta\). If the angle \(\theta\) is \(90^\circ\), then \(\cos(90^\circ) = 0\). This makes the entire dot product zero!

Example: Show that the vectors \(\vec{u} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k}\) and \(\vec{v} = 4\mathbf{i} + 5\mathbf{j} - 2\mathbf{k}\) are orthogonal.

We just need to calculate their dot product.
\( \vec{u} \cdot \vec{v} = (3)(4) + (-2)(5) + (1)(-2) = 12 - 10 - 2 = 0 \)
Since the dot product is \(0\), the vectors are orthogonal.

Key Takeaway

Parallel Check: Is \(\vec{a} = k\vec{b}\)? (Are the components proportional?)
Perpendicular Check: Is \(\vec{a} \cdot \vec{b} = 0\)? (Is the dot product zero?)

3. Finding the Angle Between Two Vectors

We just used the dot product to check for \(90^\circ\) angles, but we can use it to find ANY angle between two vectors. This is one of the most powerful applications of the dot product.

By rearranging the dot product formula, we get:

\( \cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} \)

Step-by-Step Guide to Finding the Angle

Step 1: Calculate the dot product (\(\vec{a} \cdot \vec{b}\)).
Step 2: Calculate the magnitude of each vector (\(|\vec{a}|\) and \(|\vec{b}|\)).
Step 3: Plug these three values into the formula to find \(\cos\theta\).
Step 4: Use the inverse cosine function (\(\theta = \arccos(\dots)\)) on your calculator to find the angle \(\theta\).

Example: Find the angle between \(\vec{a} = 2\mathbf{i} + 2\mathbf{j} - \mathbf{k}\) and \(\vec{b} = 6\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}\).

Step 1: Dot Product
\( \vec{a} \cdot \vec{b} = (2)(6) + (2)(-3) + (-1)(2) = 12 - 6 - 2 = 4 \)

Step 2: Magnitudes
\( |\vec{a}| = \sqrt{2^2 + 2^2 + (-1)^2} = \sqrt{4+4+1} = \sqrt{9} = 3 \)
\( |\vec{b}| = \sqrt{6^2 + (-3)^2 + 2^2} = \sqrt{36+9+4} = \sqrt{49} = 7 \)

Step 3: Formula
\( \cos\theta = \frac{4}{(3)(7)} = \frac{4}{21} \)

Step 4: Find Angle
\( \theta = \arccos\left(\frac{4}{21}\right) \approx 79.0^\circ \)
The angle between the two vectors is about \(79.0^\circ\).

Quick Review Box

Angle Formula: \( \theta = \arccos\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right) \)

4. Projection of a Vector

What is a projection? Imagine the sun is directly overhead. Your shadow on the ground is your "projection" onto the ground. In vectors, we do something similar. The projection of vector \(\vec{a}\) onto vector \(\vec{b}\) is like finding the "shadow" that \(\vec{a}\) casts along the direction of \(\vec{b}\).

There are two types of projection: scalar and vector.

Scalar Projection (The Length of the Shadow)

This tells you the signed magnitude or length of the shadow along the direction of \(\vec{b}\). It's a scalar value.

The formula for the scalar projection of \(\vec{a}\) onto \(\vec{b}\) is:

\( \text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \)

Vector Projection (The Shadow as a Vector)

This gives you the actual shadow vector. It has both magnitude and direction.

To get the vector, we take the length (scalar projection) and multiply it by the unit vector in the direction of \(\vec{b}\) (which is \(\frac{\vec{b}}{|\vec{b}|}\)).

The formula for the vector projection of \(\vec{a}\) onto \(\vec{b}\) is:

\( \text{proj}_{\vec{b}}\vec{a} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\right) \frac{\vec{b}}{|\vec{b}|} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b} \)

Common Mistake Alert!

When projecting \(\vec{a}\) onto \(\vec{b}\), the vector in the denominator is always \(\vec{b}\). The vector you're projecting onto determines the direction and appears in the denominator.

Example: Find the scalar and vector projection of \(\vec{a} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) onto \(\vec{b} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\).

Step 1: Find the dot product and magnitude of \(\vec{b}\).
\( \vec{a} \cdot \vec{b} = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32 \)
\( |\vec{b}| = \sqrt{4^2 + 5^2 + 6^2} = \sqrt{16 + 25 + 36} = \sqrt{77} \)

Step 2: Calculate Scalar Projection.
\( \text{Scalar Projection} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \frac{32}{\sqrt{77}} \)

Step 3: Calculate Vector Projection.
We also need \(|\vec{b}|^2 = 77\).
\( \text{Vector Projection} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b} = \frac{32}{77}(4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}) = \frac{128}{77}\mathbf{i} + \frac{160}{77}\mathbf{j} + \frac{192}{77}\mathbf{k} \)

Key Takeaway

Projection is like finding a vector's shadow.

Scalar Projection: \(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\)
Vector Projection: \(\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}\)

5. Area of Parallelograms and Triangles

We can use the cross product to find areas in 3D space directly.

Area of a Parallelogram

The magnitude of the cross product of two non-parallel vectors \(\vec{a}\) and \(\vec{b}\) spanning a parallelogram gives its area:

\( \text{Area of Parallelogram} = |\vec{a} \times \vec{b}| \)

Area of a Triangle

Since a triangle with adjacent sides \(\vec{a}\) and \(\vec{b}\) is half of the parallelogram formed by those vectors, its area is:

\( \text{Area of Triangle} = \frac{1}{2} |\vec{a} \times \vec{b}| \)

Example: Find the area of the triangle with vertices \(P(1, 1, 1)\), \(Q(2, 3, 4)\), and \(R(4, 3, 2)\).

Step 1: Find two vectors with a common vertex (e.g. \(P\)).
\( \vec{PQ} = (2-1)\mathbf{i} + (3-1)\mathbf{j} + (4-1)\mathbf{k} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \)
\( \vec{PR} = (4-1)\mathbf{i} + (3-1)\mathbf{j} + (2-1)\mathbf{k} = 3\mathbf{i} + 2\mathbf{j} + \mathbf{k} \)

Step 2: Calculate the cross product, \(\vec{PQ} \times \vec{PR}\).
\( \vec{PQ} \times \vec{PR} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 3 \\ 3 & 2 & 1 \end{vmatrix} \)
\( = \mathbf{i}(2\cdot 1 - 3\cdot 2) - \mathbf{j}(1\cdot 1 - 3\cdot 3) + \mathbf{k}(1\cdot 2 - 2\cdot 3) = -4\mathbf{i} + 8\mathbf{j} - 4\mathbf{k} \)

Step 3: Find the magnitude.
\( |-4\mathbf{i} + 8\mathbf{j} - 4\mathbf{k}| = \sqrt{(-4)^2 + 8^2 + (-4)^2} = \sqrt{16 + 64 + 16} = \sqrt{96} = 4\sqrt{6} \)

Step 4: Divide by 2.
\( \text{Area} = \frac{1}{2}(4\sqrt{6}) = 2\sqrt{6} \)

6. Scalar Triple Product and Volumes

The scalar triple product combines the dot product and cross product of three vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\):

\( \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \)

Volume of a Parallelepiped

If three coterminous edges of a parallelepiped are represented by vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), its volume \(V\) is given by the absolute value of their scalar triple product:

\( V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \)

Volume of a Tetrahedron

A tetrahedron with coterminous edges \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) has a volume equal to \(\frac{1}{6}\) of the corresponding parallelepiped:

\( V = \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| \)

Coplanar Vectors

Three non-zero vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) lie on the same plane (are coplanar) if and only if the volume of the solid they form is zero:

\( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \)

Example: Check whether the vectors \(\vec{a} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\), \(\vec{b} = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}\), and \(\vec{c} = 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}\) are coplanar.

Evaluate the scalar triple product:
\( \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{vmatrix} \)
\( = 1(15 - 16) - 2(10 - 12) + 3(8 - 9) = 1(-1) - 2(-2) + 3(-1) = -1 + 4 - 3 = 0 \)
Since the scalar triple product is \(0\), the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar.

Key Summary Table

Area of Triangle: \( \frac{1}{2} |\vec{a} \times \vec{b}| \)
Volume of Parallelepiped: \( |\vec{a} \cdot (\vec{b} \times \vec{c})| \)
Volume of Tetrahedron: \( \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| \)
Coplanarity Condition: \( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \)

References

Mathematics Education - Curriculum Documents — Education Bureau (EDB), HKSAR
Ongoing Renewal of the School Curriculum — Education Bureau (EDB), HKSAR

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the section formula for a point dividing a line segment internally in the ratio \(m : n\)?

\( \vec{p} = \frac{n\vec{a} + m\vec{b}}{m+n} \)

What is the section formula for a point dividing a line segment externally in the ratio \(m : n\)?

\( \vec{p} = \frac{-n\vec{a} + m\vec{b}}{m-n} \quad (m \neq n) \)

What is the condition for two non-zero vectors to be parallel?

Two non-zero vectors \(\vec{a}\) and \(\vec{b}\) are parallel if and only if \(\vec{a} = k\vec{b}\) for some non-zero scalar \(k\).

What is the condition for two non-zero vectors to be orthogonal (perpendicular)?

Two non-zero vectors \(\vec{a}\) and \(\vec{b}\) are orthogonal if and only if \(\vec{a} \cdot \vec{b} = 0\).

How can you find the angle \(\theta\) between two non-zero vectors \(\vec{a}\) and \(\vec{b}\)?

\( \theta = \arccos\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right) \)

What are the formulas for the scalar projection and vector projection of \(\vec{a}\) onto \(\vec{b}\)?

Scalar projection: \( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \); Vector projection: \( \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b} \)

How is the cross product used to find the area of a parallelogram and a triangle formed by vectors \(\vec{a}\) and \(\vec{b}\)?

\( \text{Area of Parallelogram} = |\vec{a} \times \vec{b}| \) and \( \text{Area of Triangle} = \frac{1}{2}|\vec{a} \times \vec{b}| \)

How is the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c})\) calculated using coordinates?

\( \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \)

What are the vector formulas for the volume of a parallelepiped and a tetrahedron with coterminous edges \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\)?

\( V_{\text{parallelepiped}} = |\vec{a} \cdot (\vec{b} \times \vec{c})| \) and \( V_{\text{tetrahedron}} = \frac{1}{6}|\vec{a} \cdot (\vec{b} \times \vec{c})| \)

What is the scalar triple product condition for three non-zero vectors to be coplanar?

\( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \)

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