MTH 103 Elementary Mathematics III: Vectors, Study Guide

MTH 103 Elementary Mathematics III (Vectors, Geometry and Dynamics) is the third 100 level mathematics course in the National Universities Commission's Core Curriculum and Minimum Academic Standards (CCMAS). It is a compulsory 2-unit course in the B.Sc. Mathematics programme. It brings together vectors, plane coordinate geometry and the mechanics of a particle, which makes it the first taste of multivariable calculus for most students. This guide maps the MTH 103 topics to free practice.

What MTH 103 covers

According to the CCMAS outline, MTH 103 begins with the geometric representation of vectors in one to three dimensions, components and direction cosines. It then covers addition and scalar multiplication of vectors, linear independence, and the scalar and vector products of two vectors, before differentiating and integrating vectors with respect to a scalar variable.

The geometry strand covers two-dimensional coordinate geometry: straight lines, circles, the parabola, ellipse and hyperbola, and tangents and normals. The dynamics strand studies the kinematics of a particle, components of velocity and acceleration in a plane, force and momentum, motion under gravity, projectiles and resisted vertical motion, the elastic string and simple pendulum, and impulse and impact of smooth spheres.

The vector and curve topics line up with the opening unit of a standard Calculus 3 course. The mechanics topics belong to applied mathematics, and we do not have lessons on impulse, impact or pendulums.

MTH 103 topic map

How MTH 103 fits the CCMAS sequence

MTH 103 uses MTH 102 calculus whenever it differentiates a position vector or finds the slope of a tangent to a conic. MTH 210 picks up the vector thread at 200 level.

Example problems at the MTH 103 level

These are original practice problems.

1. Angle between vectors. Let \mathbf{a} = (1, 2, 2) and \mathbf{b} = (3, 0, 4). Then |\mathbf{a}| = 3, |\mathbf{b}| = 5 and \mathbf{a}\cdot\mathbf{b} = 3 + 0 + 8 = 11, so

The direction cosines of \mathbf{a} are \frac{1}{3}, \frac{2}{3}, \frac{2}{3}.

2. A vector product.

Check: (8, 2, -6)\cdot(1, 2, 2) = 8 + 4 - 12 = 0, so the result is perpendicular to \mathbf{a} as it should be.

3. A projectile. With g = 10\text{ m/s}^2, a particle has position \mathbf{r}(t) = (20t,\; 20t - 5t^2). Its velocity is \mathbf{v}(t) = (20,\; 20 - 10t). The vertical component is zero at t = 2, so the greatest height is 20(2) - 5(2)^2 = 20 m.

4. A tangent to a circle. On x^2 + y^2 = 25 at (3, 4), implicit differentiation gives \frac{dy}{dx} = -\frac{x}{y} = -\frac{3}{4}, so the tangent is 3x + 4y = 25.

How to do well in MTH 103

• Make the vector products automatic. Drill the dot product and the cross product until you can do them without notes.
• Draw before you calculate. Every geometry and dynamics question is easier with a labelled diagram and clearly chosen axes.
• Lean on MTH 102. Velocity is the derivative of position and displacement is the integral of velocity, so keep the formula sheet handy.
• Check your answers. A cross product should be perpendicular to both inputs, and a tangent line should pass through the given point. Our common calculus mistakes list shows other quick checks, and practice lets you test yourself on fresh problems.