Introduction to Vectors

Introduction to vectors is one of the two biggest focus areas in Extension 1: it was worth 14 marks on NESA's sample paper for the 2024 syllabus and 11–14 marks in most recent HSC papers. Expect multiple choice questions on the scalar product and a long Section II question on motion or projectiles. The 2024 syllabus adds 3D vectors and relative velocity.

• Representation: vectors as directed line segments, notation, position vectors, equal vectors.
• 2D and 3D vectors: points in space, the distance formula, coordinate planes, unit vectors $\underset{\sim }{i}$, $\underset{\sim }{j}$, $\underset{\sim }{k}$, component and column form.
• Operations: scalar multiples, parallel vectors, addition and subtraction, magnitude, unit vectors.
• The scalar product: algebraic and geometric forms, angles, perpendicular and parallel tests, projections.
• Motion in two dimensions: position, velocity and acceleration vectors, parametric paths, constant velocity and relative velocity (crosswind or current).
• Projectile motion: derive the equations of motion from $\underset{\sim }{a}=-g\underset{\sim }{j}$, and find time of flight, maximum height, range and impact velocity.

Vector equations of lines and the vector (cross) product are not in this course.

A vector has magnitude and direction; a scalar is just a real number. In handwriting, vectors are written with a tilde underneath, $\underset{\sim }{a}$, or as $\overrightarrow{AB}$ for the vector from $A$ to $B$. Vectors with the same magnitude and direction are equal, wherever they are drawn. A position vector has its tail at the origin.

In 3D, the $z$-axis is added. Points on the $xy$-plane have $z=0$, on the $xz$-plane $y=0$ and on the $yz$-plane $x=0$; these are the equations of the coordinate planes.

For points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$, $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$, so its components are "end minus start", and $|\overrightarrow{AB}|$ is the distance $AB$. Drop the $z$ terms for 2D.

Operations

• Scalar multiple: $\lambda \underset{\sim }{a}$ multiplies every component by $\lambda$; if $\lambda <0$ the direction reverses.
• Parallel: non-zero $\underset{\sim }{a}$ and $\underset{\sim }{b}$ are parallel if $\underset{\sim }{a}=\lambda \underset{\sim }{b}$ for some $\lambda \ne 0$.
• Add or subtract component by component; geometrically, use the triangle or parallelogram rule.
• Unit vector: $\underset{\sim }{\hat{a}}=\frac{1}{|\underset{\sim }{a}|}\underset{\sim }{a}$ has magnitude 1 in the direction of $\underset{\sim }{a}$.

The scalar (dot) product has two equivalent forms. In components (3D shown; drop $z$ for 2D):

Geometrically, with $\theta$ the angle between the vectors ($0\le \theta \le \pi$):

Consequences you must know and be able to prove:

• $\underset{\sim }{a}\cdot \underset{\sim }{a}=|\underset{\sim }{a}{|}^2$, which is very useful in proofs.
• Perpendicular (non-zero vectors): $\underset{\sim }{a}\cdot \underset{\sim }{b}=0$.
• Parallel: $\underset{\sim }{a}\cdot \underset{\sim }{b}=\pm |\underset{\sim }{a}||\underset{\sim }{b}|$.
• If $|\underset{\sim }{a}|=|\underset{\sim }{b}|$ then $\underset{\sim }{a}+\underset{\sim }{b}$ and $\underset{\sim }{a}-\underset{\sim }{b}$ are perpendicular, because their dot product is $|\underset{\sim }{a}{|}^2-|\underset{\sim }{b}{|}^2=0$.

The projection of $\underset{\sim }{a}$ onto $\underset{\sim }{b}$ is the vector "shadow" of $\underset{\sim }{a}$ along $\underset{\sim }{b}$. The part of $\underset{\sim }{a}$ perpendicular to $\underset{\sim }{b}$ is $\underset{\sim }{a}-{\text{proj}}_{\underset{\sim }{b}}\underset{\sim }{a}$.

To see why: the projection is $\lambda \underset{\sim }{b}$ with $\underset{\sim }{a}-\lambda \underset{\sim }{b}$ perpendicular to $\underset{\sim }{b}$, so

Worked example 1: angle and projection in 3D

Let $\underset{\sim }{a}=\underset{\sim }{i}+2\underset{\sim }{j}+2\underset{\sim }{k}$ and $\underset{\sim }{b}=2\underset{\sim }{i}+3\underset{\sim }{j}+6\underset{\sim }{k}$. Find the angle between them, ${\text{proj}}_{\underset{\sim }{b}}\underset{\sim }{a}$, and the component of $\underset{\sim }{a}$ perpendicular to $\underset{\sim }{b}$.

Check: the dot product of this with $\underset{\sim }{b}$ is $\frac{18+114-132}{49}=0$, so it is perpendicular to $\underset{\sim }{b}$ as it should be.

A particle's position at time $t$ is $\underset{\sim }{r}(t)=x(t)\underset{\sim }{i}+y(t)\underset{\sim }{j}$. The pair $x(t)$, $y(t)$ are parametric equations of the path; eliminate $t$ to get the Cartesian equation (a line, parabola or circle in this course). For a circle, use ${\cos }^{2}t+{\sin }^{2}t=1$.

Differentiate to get velocity and acceleration, and integrate (using initial conditions for each component) to go back:

Speed is $|\underset{\sim }{v}|$; the distance between two positions is the magnitude of the displacement vector.

Relative velocity

When an object moves through a medium that is itself moving (a plane in wind, a boat in a current), its actual velocity is the sum of its velocity relative to the medium and the medium's velocity. Draw the vector triangle and describe the direction as a bearing or as an angle from a stated direction.

For example, a plane heads due north at 400 km/h in a wind blowing towards the east at 60 km/h. Its ground velocity is $60\underset{\sim }{i}+400\underset{\sim }{j}$, so its ground speed is $\sqrt{60^2+400^2}\approx 404.5$ km/h on a bearing of ${\tan }^{-1}\frac{60}{400}\approx 008.5^{\circ }$ (rounded to 1 decimal place).

Near the Earth's surface, with air resistance ignored, the only force is gravity, so $\underset{\sim }{a}=-g\underset{\sim }{j}$. Integrate twice, using the initial velocity $V\cos \theta \underset{\sim }{i}+V\sin \theta \underset{\sim }{j}$ and the initial position. You may be asked to derive these, so practise doing it from $\underset{\sim }{a}$, not just quoting the result.

That is for launch from the origin; add the starting height to the $\underset{\sim }{j}$ component otherwise.

• Maximum height: the vertical velocity is 0.
• Time of flight: the vertical position equals the landing height (often 0).
• Range: the horizontal position at the time of flight.
• Impact velocity: substitute the time of flight into $\underset{\sim }{v}$; give the speed and the angle to the horizontal.

Worked example 2: a projectile from a platform

A ball is launched from a platform 20 m above level ground with speed 25 m/s at angle $\theta$ above the horizontal, where $\tan \theta =\frac{3}{4}$. Take $g=10$ m/s². Find its maximum height, horizontal range, impact speed and the Cartesian equation of its path.

Since $\sin \theta =\frac{3}{5}$ and $\cos \theta =\frac{4}{5}$, the initial velocity is $20\underset{\sim }{i}+15\underset{\sim }{j}$. Integrating $\underset{\sim }{a}=-10\underset{\sim }{j}$ with $\underset{\sim }{r}(0)=20\underset{\sim }{j}$:

Maximum height: $15-10t=0$ gives $t=1.5$, and $y=20+22.5-11.25=31.25$ m.

Range: $20+15t-5t^2=0$ gives $t^2-3t-4=0$, so $(t-4)(t+1)=0$ and $t=4$ s (reject $t=-1$). The range is $20\times 4=80$ m.

Impact: $\underset{\sim }{v}(4)=20\underset{\sim }{i}-25\underset{\sim }{j}$, so the speed is $\sqrt{1025}\approx 32.0$ m/s at ${\tan }^{-1}\frac{25}{20}\approx 51.3^{\circ }$ below the horizontal.

Path: $t=\frac{x}{20}$, so

• Subtracting the wrong way round: $\overrightarrow{AB}=\underset{\sim }{b}-\underset{\sim }{a}$, end minus start.
• Using the scalar projection (a number) when the question asks for the vector projection, or dividing by $|\underset{\sim }{b}|$ instead of $|\underset{\sim }{b}{|}^2$.
• Finding the angle between vectors without checking that $\theta$ is between $0^{\circ }$ and $180^{\circ }$; a negative dot product means an obtuse angle.
• Forgetting the constant of integration in each component, or using the wrong initial position when launching from a height.
• Giving impact speed but not direction (or vice versa) when asked for the velocity.
• In relative velocity, adding the wind's "from" direction instead of the direction it blows towards.

• Derive the equations of motion means start from $\underset{\sim }{a}=-g\underset{\sim }{j}$, integrate, and use initial conditions. Quoting the final formulas earns little.
• Show that vectors are perpendicular: calculate the dot product and say it equals 0. For parallel, find $\lambda$ explicitly.
• Vector proofs in geometry (rhombus diagonals, angles in a semicircle) usually come down to writing each side in terms of two base vectors and using $\underset{\sim }{a}\cdot \underset{\sim }{a}=|\underset{\sim }{a}{|}^2$.
• When the angle or speed of projection is unknown, write the conditions you know as equations in $V$ and $\theta$ and solve them simultaneously; ${\sin }^2\theta +{\cos }^2\theta =1$ and $\tan \theta =\frac{\sin \theta }{\cos \theta }$ are the usual tools.
• Round only at the end and state units on every answer.

• Advanced, Applications of Calculus (MA12-AC): motion in a straight line; vector motion applies the same calculus to each component.
• Further calculus skills (ME12-CS): parametric equations and their derivatives give the gradient of a path.
• Further trigonometry (ME11-TR): 3D diagrams, the cosine rule and exact values appear in vector and projectile problems.
• Inverse trigonometric functions (ME12-IT): angles between vectors and directions of motion use ${\cos }^{-1}$ and ${\tan }^{-1}$.