In VCE Specialist Mathematics, vectors provide a powerful framework for describing and solving problems involving lines and planes in three-dimensional space. This area of study bridges the gap between algebraic manipulation and geometric interpretation.
A vector in three dimensions is represented as \(\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\), where \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are unit vectors in the directions of the \(x, y,\) and \(z\) axes respectively.
The scalar product is used to find the angle between vectors and to test for perpendicularity.
\$\(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\mathbf{a}||\mathbf{b}|\cos(\theta)\)\$
* If \(\mathbf{a} \cdot \mathbf{b} = 0\), then \(\mathbf{a} \perp \mathbf{b}\) (provided \(\mathbf{a}, \mathbf{b} \neq \mathbf{0}\)).
The vector product results in a vector that is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).
\$\(\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}\)\$
* Magnitude: \(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin(\theta)\).
* Geometric use: The area of a triangle with sides defined by vectors \(\mathbf{a}\) and \(\mathbf{b}\) is \(\frac{1}{2}|\mathbf{a} \times \mathbf{b}|\).
EXAM TIP: Use the dot product to find angles and the cross product to find a vector perpendicular to a plane. If a question asks to show two vectors are perpendicular, always show that \(\mathbf{a} \cdot \mathbf{b} = 0\).
A line in 3D space is uniquely determined by a point on the line and a direction vector.
Let \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}\) be the position vector of a point on the line, and \(\mathbf{d} = d_1\mathbf{i} + d_2\mathbf{j} + d_3\mathbf{k}\) be the direction vector parallel to the line.
| Form | Equation |
|---|---|
| Vector Form | \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}, \lambda \in \mathbb{R}\) |
| Parametric Form | \(x = a_1 + \lambda d_1, \quad y = a_2 + \lambda d_2, \quad z = a_3 + \lambda d_3\) |
| Cartesian Form | \(\frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3}\) |
COMMON MISTAKE: When converting to Cartesian form, ensure the coefficients of \(x, y,\) and \(z\) are exactly \(1\). For example, if you have \(\frac{2x-4}{6}\), rewrite it as \(\frac{x-2}{3}\) before identifying the direction vector component.
A plane is defined by a point \(\mathbf{a}\) on the plane and a normal vector \(\mathbf{n}\) (a vector perpendicular to the surface).
The set of all points \(\mathbf{r}\) on the plane satisfies:
\$\((\mathbf{r} - \mathbf{a}) \cdot \mathbf{n} = 0 \implies \mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}\)\$
Where \(\mathbf{a} \cdot \mathbf{n}\) results in a constant \(k\).
If \(\mathbf{n} = n_1\mathbf{i} + n_2\mathbf{j} + n_3\mathbf{k}\), the equation is:
\$\(n_1x + n_2y + n_3z = k\)\$
KEY TAKEAWAY: The coefficients of \(x, y,\) and \(z\) in the Cartesian equation of a plane are the components of the normal vector \(\mathbf{n}\).
| Target | Method | Formula |
|---|---|---|
| Between two lines | Use direction vectors \(\mathbf{d_1}, \mathbf{d_2}\) | \$\cos(\theta) = \frac{ |
| Between two planes | Use normal vectors \(\mathbf{n_1}, \mathbf{n_2}\) | \$\cos(\theta) = \frac{ |
| Between line and plane | Use direction \(\mathbf{d}\) and normal \(\mathbf{n}\) | \$\sin(\theta) = \frac{ |
VCAA FOCUS: The angle between a line and a plane is a frequent exam question. Remember to use sine instead of cosine if you are using the normal vector of the plane, or calculate the angle with the normal and subtract from \(90^\circ\).
Vectors can be used to prove geometric properties:
1. Collinearity: Three points \(A, B, C\) are collinear if \(\vec{AB} = k\vec{BC}\) for some scalar \(k\).
2. Coplanarity: Four points are coplanar if the volume of the parallelepiped they form is zero, or if one vector can be expressed as a linear combination of two others: \(\mathbf{c} = \alpha\mathbf{a} + \beta\mathbf{b}\).
3. Isosceles Triangles: Show that two side vectors have equal magnitudes (e.g., \(|\vec{AB}| = |\vec{AC}|\)).
4. Intersection: To find the intersection of a line and a plane, substitute the parametric expressions for \(x, y, z\) from the line into the Cartesian equation of the plane and solve for \(\lambda\).
REMEMBER: To show that a line is perpendicular to a plane, the direction vector of the line \(\mathbf{d}\) must be parallel to the normal vector of the plane \(\mathbf{n}\) (i.e., \(\mathbf{d} = k\mathbf{n}\)). To show a line is parallel to a plane, \(\mathbf{d} \cdot \mathbf{n} = 0\).
