Algebraic Operations with Vectors

In VCE Specialist Mathematics, a vector is defined as a quantity that possesses both magnitude and direction. This distinguishes it from a scalar, which has magnitude only (e.g., mass, time, or speed).

1. Vector Notation and Representation

Vectors can be represented geometrically as directed line segments or algebraically using components.

• Geometric Notation: A vector from point \(O\) to point \(A\) is denoted as \(\vec{OA}\).
• Algebraic Notation: Vectors are typically written as bold lowercase letters \(\mathbf{a}\) or underlined letters \(\underline{a}\).
• Component Form (2D and 3D):
  • In 2D: \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\)
  • In 3D: \(\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\)
  • Where \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are unit vectors in the directions of the positive \(x, y,\) and \(z\) axes respectively.

KEY TAKEAWAY: A vector is not fixed in space; two vectors are considered equal if they have the same magnitude and the same direction, regardless of their starting points.

2. Vector Addition

Vector addition can be understood both geometrically and algebraically.

Geometric Addition

1. Triangle Rule: To find \(\mathbf{a} + \mathbf{b}\), place the “tail” of vector \(\mathbf{b}\) at the “head” of vector \(\mathbf{a}\). The resultant vector is the shortcut from the tail of \(\mathbf{a}\) to the head of \(\mathbf{b}\).
2. Parallelogram Rule: If two vectors \(\mathbf{a}\) and \(\mathbf{b}\) start from the same point, they form two sides of a parallelogram. The resultant \(\mathbf{a} + \mathbf{b}\) is the diagonal starting from that same point.

Algebraic Addition (Component Form)

To add vectors algebraically, add their corresponding components:
If \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}\), then:
\$\(\mathbf{a} + \mathbf{b} = (a_1 + b_1)\mathbf{i} + (a_2 + b_2)\mathbf{j} + (a_3 + b_3)\mathbf{k}\)\$

Properties of Addition

• Commutative: \(\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}\)
• Associative: \((\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})\)
• Identity: \(\mathbf{a} + \mathbf{0} = \mathbf{a}\) (where \(\mathbf{0}\) is the zero vector).

EXAM TIP: When a question involves a closed geometric figure (like a triangle or hexagon), the sum of vectors forming a closed loop returning to the start point is always the zero vector \(\mathbf{0}\).

3. Vector Subtraction

Subtraction is defined as the addition of a negative vector. The negative vector \(-\mathbf{b}\) has the same magnitude as \(\mathbf{b}\) but points in the opposite direction.

Geometric Interpretation

If \(\mathbf{a}\) and \(\mathbf{b}\) are drawn from the same origin \(O\), such that \(\vec{OA} = \mathbf{a}\) and \(\vec{OB} = \mathbf{b}\), then the vector \(\mathbf{a} - \mathbf{b}\) is the vector \(\vec{BA}\) (the vector pointing from the head of \(\mathbf{b}\) to the head of \(\mathbf{a}\)).

Algebraic Subtraction

Subtract corresponding components:
\$\(\mathbf{a} - \mathbf{b} = (a_1 - b_1)\mathbf{i} + (a_2 - b_2)\mathbf{j} + (a_3 - b_3)\mathbf{k}\)\$

COMMON MISTAKE: Students often reverse the direction of the subtraction vector. Remember: \(\mathbf{a} - \mathbf{b}\) points towards \(\mathbf{a}\). A helpful mnemonic is “Final minus Initial” (\(\vec{BA} = \text{position } A - \text{position } B\)).

4. Scalar Multiplication

Scalar multiplication involves multiplying a vector \(\mathbf{a}\) by a real number \(k\) (a scalar).

• Magnitude: The magnitude of \(k\mathbf{a}\) is \(|k|\) times the magnitude of \(\mathbf{a}\).
• Direction:
  • If \(k > 0\), the direction remains the same.
  • If \(k < 0\), the direction is reversed.
  • If \(k = 0\), the result is the zero vector \(\mathbf{0}\).

Parallelism

Two non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel if and only if one is a scalar multiple of the other:
\$\(\mathbf{a} \parallel \mathbf{b} \iff \mathbf{a} = k\mathbf{b} \text{ for some } k \in \mathbb{R} \setminus \{0\}\)\$

Unit Vectors

A unit vector is a vector with a magnitude of 1. To find a unit vector \(\hat{\mathbf{a}}\) in the direction of \(\mathbf{a}\):
\$\(\hat{\mathbf{a}} = \frac{1}{|\mathbf{a}|}\mathbf{a}\)\$
where \(|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\).

STUDY HINT: Scalar multiplication is distributive over vector addition: \(k(\mathbf{a} + \mathbf{b}) = k\mathbf{a} + k\mathbf{b}\). This is a fundamental tool for simplifying algebraic vector expressions.

5. Linear Combinations

A vector \(\mathbf{r}\) is a linear combination of vectors \(\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\) if it can be expressed in the form:
\$\(\mathbf{r} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_n\mathbf{v}_n\)\$
where \(c_1, c_2, \dots, c_n\) are scalar coefficients.

Linear Independence and Dependence

• Linearly Independent: A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others.
  • In 2D: Two vectors are linearly independent if they are not parallel.
• Linearly Dependent: A set of vectors is linearly dependent if at least one vector can be written as a linear combination of the others.
  • Three vectors in 2D are always linearly dependent.
  • Four vectors in 3D are always linearly dependent.

Solving for Coefficients

If a vector \(\mathbf{p}\) is a linear combination of \(\mathbf{a}\) and \(\mathbf{b}\), we set up the equation:
\$\(\mathbf{p} = m\mathbf{a} + n\mathbf{b}\)\$
By equating the \(\mathbf{i}, \mathbf{j}\) and \(\mathbf{k}\) components, we create a system of simultaneous equations to solve for the scalars \(m\) and \(n\).

VCAA FOCUS: Many exam questions require you to prove that three points \(A, B,\) and \(C\) are collinear. This is achieved by showing that the vectors \(\vec{AB}\) and \(\vec{BC}\) (or \(\vec{AC}\)) are parallel (scalar multiples) and share a common point \(B\).

Summary Table of Algebraic Operations

REMEMBER: The zero vector \(\mathbf{0} = 0\mathbf{i} + 0\mathbf{j} + 0\mathbf{k}\) is the only vector with a magnitude of 0 and no defined direction. It is essential for solving vector equations where all terms cancel out.