H2 Maths 3.1 - Basic Vector Properties: Magnitude, Unit Vector, and Parallel Vectors

Step 1 - Read a vector from components

Start with the vector a equals two i minus j plus two k.

Step 1 - Read a vector from components

In column-vector form, its components are two, negative one, and two.

Step 1 - Read a vector from components

Each component tells you how far the vector moves in the x, y, and z directions.

Step 1 - Read a vector from components

So before doing any vector calculation, translate between i j k form and column-vector form cleanly.

Step 2 - Find the magnitude

The magnitude of a vector is its length.

Step 2 - Find the magnitude

Use Pythagoras in three dimensions: square each component, add, then square root.

Step 2 - Find the magnitude

For this vector, two squared plus negative one squared plus two squared gives nine, so the magnitude is three.

Step 3 - Find a unit vector

A unit vector has length one and points in the same direction.

Step 3 - Find a unit vector

To form it, divide the vector by its own magnitude.

Step 3 - Find a unit vector

Since the magnitude of a is three, the unit vector in the direction of a is one third of a.

Step 4 - Test whether two vectors are parallel

Now compare a with b equals negative four, two, negative four.

Step 4 - Test whether two vectors are parallel

Vectors are parallel when one is a scalar multiple of the other.

Step 4 - Test whether two vectors are parallel

Here b is negative two times a, so they are parallel but point in opposite directions.

Step 5 - Common exam pitfall

The zero vector is special.

Step 5 - Common exam pitfall

It has magnitude zero, so you cannot divide by its magnitude to form a unit vector.

Step 5 - Common exam pitfall

Also remember that a position vector locates a point from the origin, while a direction vector only tells you direction.

Step 5 - Common exam pitfall

That distinction matters when writing lines and planes.