Introduction to Column Vectors

• Independence of Components: The horizontal and vertical movements of a vector are independent. Operations performed on the $x$-component do not affect the $y$-component.

• Vector Equality: Two column vectors are considered equal if and only if their corresponding components are identical. This means $\begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} c \\ d \end{pmatrix}$ implies $a=c$ and $b=d$.

• Magnitude and Direction: Every vector possesses both a size (magnitude) and a specific orientation (direction). If the direction is reversed, the signs of both components are flipped.

Vector Addition and Subtraction

• To add two vectors, simply add the corresponding top numbers together and the corresponding bottom numbers together: $\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}$
• To subtract vectors, subtract the components of the second vector from the first: $\begin{pmatrix} a \\ b \end{pmatrix} - \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a-c \\ b-d \end{pmatrix}$

Scalar Multiplication

• A scalar is a single real number (not a vector) that scales the magnitude of a vector.
• To multiply a vector by a scalar $k$, multiply every component by that scalar: $k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}$
• If $k > 1$, the vector is stretched; if $0 < k < 1$, it is compressed; if $k$ is negative, the direction is reversed.

• It is vital to distinguish between a position coordinate and a displacement vector.

• Vector vs. Scalar: A vector has both magnitude and direction (e.g., velocity), while a scalar has only magnitude (e.g., speed or time).

• Check the Signs: Always verify the direction of movement. A common mistake is forgetting that 'down' or 'left' requires a negative sign in the column vector.

• Notation Awareness: In exams, vectors are often printed in bold (e.g., a). When writing by hand, you should underline the letter (e.g., $\underline{a}$) to indicate it is a vector.

• Visual Verification: If you calculate a resultant vector, do a quick mental sketch. If your vector is $\begin{pmatrix} 5 \\ -2 \end{pmatrix}$, your arrow should point generally 'down and to the right'.

• Order of Operations: When solving expressions like $2\mathbf{a} + 3\mathbf{b}$, perform the scalar multiplication first for each vector before adding them together.

• Swapping Components: Students often accidentally put the $y$ (vertical) component on top and the $x$ (horizontal) component on the bottom. Remember: 'x is across, y is up/down'.

• The Zero Vector: The vector $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$ represents no movement. It is still a vector, not just the number zero.

• Confusing Addition with Coordinates: Do not put a comma between the numbers in a column vector; this is a notation error that can lead to confusion with coordinates.