Comprehensive Vectors Revision Notes

Getting Started with Vectors

Think of vectors as arrows that tell you exactly where to go and how far to travel. Unlike regular numbers, vectors care about direction just as much as size.

You can name vectors using the start and end points (like $\overline{AB}$) or with a single letter (usually bold or underlined). In component form, vectors look like $\binom{2}{4}$, where the top number shows horizontal movement and the bottom shows vertical movement.

Adding and subtracting vectors works just like regular arithmetic - you simply combine the corresponding components. For example, $\binom{2}{4} + \binom{-3}{1} = \binom{-1}{5}$. When multiplying by a scalar (a regular number), you multiply each component separately.

Key Point: Never try to simplify vectors like fractions - each component stays separate!

Magnitude and Position Vectors

The magnitude of a vector is its length, calculated using Pythagoras' theorem. For a 2D vector $\binom{2}{4}$, the magnitude is $\sqrt{2^2 + 4^2} = \sqrt{20} = 2\sqrt{5}$. For 3D vectors, you just add the third component squared under the square root.

Position vectors tell you exactly where a point is located from the origin (0,0). If point P is at coordinates (x,y,z), then its position vector is $\overrightarrow{OP} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$.

Here's the crucial relationship: to find vector $\overrightarrow{AB}$, you calculate $\underline{b} - \underline{a}$ (destination minus starting point). This works for any two points and is absolutely essential for exam questions.

Exam Tip: Remember that $\overrightarrow{AB} = \underline{b} - \underline{a}$ - this formula appears in virtually every vector question!

Working with Vector Examples

Let's see how these concepts work in practice. Given points A$-12, 4$ and B$5, -2$, you can write their position vectors as $\underline{a} = \begin{pmatrix} -12 \\ 4 \end{pmatrix}$ and $\underline{b} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}$.

To find $\overrightarrow{AB}$, you calculate $\underline{b} - \underline{a} = \begin{pmatrix} 5 \\ -2 \end{pmatrix} - \begin{pmatrix} -12 \\ 4 \end{pmatrix} = \begin{pmatrix} 17 \\ -6 \end{pmatrix}$.

When calculating magnitudes, be extra careful with negative numbers. For the vector $\begin{pmatrix} -1 \\ -4 \\ -7 \end{pmatrix}$, the magnitude is $\sqrt{(-1)^2 + (-4)^2 + (-7)^2} = \sqrt{66}$. Notice how the negative signs disappear when you square each component.

Practice Makes Perfect: The more you practice these calculations, the more automatic they become on exam day!

Unit Vectors and Parallel Vectors

A unit vector is simply a vector with magnitude 1 - it shows pure direction without worrying about size. To create a unit vector from any vector, divide each component by the vector's magnitude.

For example, if $\mathbf{u} = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}$ has magnitude 5, then its unit vector is $\frac{1}{5}\begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix} = \begin{pmatrix} 3/5 \\ 0 \\ 4/5 \end{pmatrix}$.

Parallel vectors are multiples of each other. If $\mathbf{u} = k\mathbf{v}$ where k is any number, then the vectors are parallel. When k is negative, they point in opposite directions but are still parallel.

Collinearity means points lie on a straight line. If $\overrightarrow{AB} = k\overrightarrow{BC}$, then points A, B, and C are collinear because the vectors are parallel and share point B.

Remember: Parallel vectors can point in opposite directions - the key is that one is a scalar multiple of the other!

Section Formula and Ratio Splitting

When a point divides a line segment in a specific ratio, you can find its position using the section formula: if Q divides AB in ratio m:n, then $q = \frac{n}{m+n}a + \frac{m}{m+n}b$.

However, there's an easier method that many students prefer. First, find vector $\overrightarrow{AB}$. Then calculate what fraction of this vector you need based on the ratio. Finally, add this to the starting position vector.

For example, if P divides AB in ratio 1:3, then P is $\frac{1}{4}$ of the way from A to B. So $\overrightarrow{AP} = \frac{1}{4}\overrightarrow{AB}$, and you can find P's coordinates by adding this to A's position.

The key is understanding what the ratio actually means - if it's 1:3, then AP is 1 part while PB is 3 parts, making the total journey 4 parts.

Pro Tip: Draw a simple diagram to visualise the ratio - it makes the calculation much clearer!

i, j, k Components

The i, j, k notation is just another way to write vectors using unit vectors in each direction. Here, i points along the x-axis, j along the y-axis, and k along the z-axis.

Converting between notations is straightforward: $2i + 5j - 3k$ becomes $\begin{pmatrix} 2 \\ 5 \\ -3 \end{pmatrix}$, and vice versa. If a component is missing (like in $5i - 2k$), it means that component is zero.

All the same rules apply - you can add, subtract, and multiply these vectors exactly as before. For instance, $(2i + j - 4k) - (-3i + 2j + k) = 5i - j - 5k$.

When calculating magnitudes, convert to component form first, then use the standard formula. The notation might look different, but the mathematics stays exactly the same.

Flexibility: Being comfortable with both notations gives you options for tackling exam questions in whatever way feels clearest!