Introduction to Column Vectors

• Magnitude and Direction: Every vector is defined by its size (magnitude) and the path it takes (direction).

• Vector vs. Scalar: A scalar is a simple numerical value (like 5 or -2) that has magnitude but no direction, whereas a vector must have both.

• Notation Standards: In printed text, vectors are usually written in bold (e.g., a), while in handwriting, they should be underlined (e.g., $\underline{a}$) to distinguish them from standard variables.

• Equality: Two column vectors are equal if and only if their corresponding top components are equal and their corresponding bottom components are equal.

• Vector Addition: To add two column vectors, simply add the corresponding top numbers and the corresponding bottom numbers independently.

Formula: \binom{a}{b} + inom{c}{d} = \binom{a+c}{b+d}

• Vector Subtraction: To subtract one vector from another, subtract the components of the second vector from the first.

Formula: \binom{a}{b} - inom{c}{d} = \binom{a-c}{b-d}

• Geometric Interpretation: Adding vectors algebraically is equivalent to placing the start of the second vector at the end of the first vector (the tip-to-tail method).

• Scaling a Vector: Multiplying a column vector by a scalar involves multiplying every component inside the vector by that scalar value.

Formula: $k \binom{x}{y} = \binom{kx}{ky}$

• Effect of Positive Scalars: If the scalar $k$ is positive, the vector's magnitude changes by a factor of $k$, but its direction remains the same.

• Effect of Negative Scalars: If the scalar $k$ is negative, the vector's magnitude is scaled, and its direction is reversed (flipped 180 degrees).

• Zero Scalar: Multiplying any vector by zero results in the zero vector $\binom{0}{0}$, which represents no movement.

Column Vectors vs. Coordinates

• It is vital to distinguish between a point's location and a vector's movement, even though they use similar numbers.

• Directional Signs: In vectors, the sign is a directional instruction. A $-3$ in the top component is not just a value; it is an instruction to move 3 units to the left.

• Check the Signs: The most common error in vector arithmetic is failing to distribute a negative sign during subtraction or scalar multiplication.

• Component Independence: Always treat the $x$ and $y$ components as completely separate equations; never mix a top number with a bottom number during calculation.

• Verification: If a question provides a grid, sketch the resulting vector to see if the direction and length look reasonable compared to your algebraic answer.

• Notation Discipline: Always underline your vectors in handwritten work to ensure clarity for the examiner, even if marks aren't explicitly deducted for omitting it.