Vector Algebra

Jamie Z 2025-05-13

Learning Goals

• Represent vectors in component form (i and j notation)
• Calculate the magnitude of a vector.
• Perform scalar multiplication on vectors.
• Add and subtract vectors.
• Determine unit vectors.
• Apply vector addition to find the position vector of a point given other points.
• Understand and apply concepts of parallel, collinear, and perpendicular vectors.
• Calculate the dot product of two vectors.
• Resolve vectors into components (scalar projection).
• Determine the angle between two vectors.

Scalar Multiplication

Multiplying a vector by a scalar k changes its magnitude. The direction remains the same if k > 0, and is reversed if k < 0.

In Cartesian form: kg = kxᵢ + kyⱼ

Vector Between Two Points

Given points A and B with position vectors a and b respectively, the vector from A to B (AB) is: \(\overrightarrow{AB} = \mathbf{b} - \mathbf{a}\)

Parallel and Collinear Vectors

Two vectors a and b are parallel if a = kb, where k is a scalar (k ≠ 0).

Two vectors are collinear if they lie along the same line. This also requires a = kb.

Dividing a line segment into a ratio

Dot Product (Scalar Product)

The dot product of two vectors a and b is denoted a ⋅ b. It's a scalar value. \(\mathbf{a} \cdot \mathbf{b} = | \mathbf{a} | | \mathbf{b} | \cos{\theta}\)

where θ is the angle between the positive directions of a and b.

Alternatively, if a = x₁ᵢ + y₁ⱼ and b = x₂ᵢ + y₂ⱼ: \(\mathbf{a} \cdot \mathbf{b} = x_1x_2 + y_1y_2\)

Perpendicular Vectors

If two vectors are perpendicular, the angle between them is 90°. Therefore their dot product is zero.

a ⋅ b = 0

Scalar Projection of a onto b

The scalar projection of vector a onto vector b gives the length of the component of a in the direction of b.

\[\]

\mathbf{a} \cdot \hat{\mathbf{b}} = |\mathbf{a}| \cos{\theta} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}

\[\]

When a light shines on a meter ruler, it projects a shadow on the floor. The scalar projection is the shadow length, i.e. scalar projection tells us how much a vector points along a certain direction.

Vector Projection

The vector projection of vector a onto vector b, gives the component of a that lies along b. It's a vector quantity.

• (a ⋅ b) : The dot product gives the amount of a that is in the direction of b.
• |b|: Dividing by |b| normalizes the projection to the length of b.
• b: Multiplying by b ensures the result is a vector pointing in the same direction as b.

Angle Between Two Vectors

Using the dot product formula, we can find the angle θ between two vectors:

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