Vectors

2 + 2 does not always equal 4 when the problem is translated into a 2 or 3 dimensional space. Vectors are quantities that have both magnitude (size) and direction. This branch of maths is fundamental to physics and engineering to represent physical quantities that have a direction.

Use these measurement worksheets to improve your skills in these areas.

The University of Colorado has some useful vector simulations, so come back and have a look at those once you've used the worksheets below.

• V1 Introduction to vectors

This page Introduces the basic concepts around vectors (which are measurable quantities with direction such as force). It introduces the concept of a scalar (which is a quantity without a direction such as heat or thermal energy).
It covers vector components (aligned along the horizontal and vertical or the 3-dimensional axes), unit vectors (a vector of size = 1), vector magnitude (or size) and how to add and subtract vectors.

• V2 Resolution of vectors

Vector quantities in combination will enhance or counteract each other, depending on their direction. An opposing force for example will lessen another force. However, if the forces (or vector quantities) do not act along exactly the same line, it is difficult to know how they will interact.
This task is easier if you can break the vectors into components which are parallel, and then find the result by adding or subtracting parallel components. This technique is very important for resolving forces and other vector quantities in physics.

• V3 Scalar product

What is a scalar product? What is a dot product? This is the result of multiplying the magnitudes of the components of two or more vectors. The result is not a vector, but a scalar (which is without direction).

• V4 Vector product

What is a vector product? What is a cross product? The vector product is a vector that is the result of multiplying the magnitudes (size) of two vectors. The magnitude is found using matrices and determinants).
The result of the cross product is another vector, and the direction is perpendicular (normal, at 90 degrees) to the plane of both the original vectors.

• V5 Projection of vectors

In ’Resolution of vectors’ we learned how to resolve vectors in two dimensions along horizontal and vertical axes. It is also possible to resolve one vector along the line of another vector (instead of along the x-y axes).
Learn how to find the projection (resolution) of one vector in the direction of a second vector.

• V6 Vector equation of a line

If you want to uniquely define a line, you need to pin it between two points in 3-dimensional space. You can also define a point with a 3-dimensional vector through it. This process uses three types of equations.
Learn how to find the vector equation, the parametric equation, and the symmetric equation of a line in three-dimensional space.

• V7 Intersecting lines in 3D

Learn how to determine if two lines in three dimensions intersect (cross each other) and, if so, what is their point of intersection?

• V8 Equation of a plane

Learn how to find the equation of a plane (a 2-dimensional space):
a) through three points or
b) given a normal (line at right angles) and a point on the plane or
c) given a parallel plane and a point on the plane.

• V9 Intersecting planes

Two 2-dimensional planes will slice through each other (unless they are parallel). Where they slice will be defined by a straight line. There will also be an angle between the two planes.
Learn how to determine the angle between two intersecting planes and the equation of the line of intersection.

• V10 Distance from a point to a plane

The shortest distance from some point in the air down to flat ground, is defined by a line straight down, sitting at right angles to the ground.
Learn how to find the perpendicular (right angle) distance from a point to a plane.

• V11 Directional derivatives

If you are on the side of a hill, the gradient depends on the direction you look. So the directional derivative is the gradient in a particular direction.
Learn how to find the directional derivative of a function of two variables f(x,y) or three variables g(x,y,z) at a point and in a given direction. This is very useful in engineering and especially computer graphics.

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