Vector Notation & Magnitude

A vector carries magnitude and direction. 0606 writes them two ways, component form $x\mathbf{i} + y\mathbf{j}$ or column form, and you must move freely between them. Vectors add componentwise, and scalar multiples scale each component.

Magnitude: Pythagoras in disguise

$|3\mathbf{i} - 4\mathbf{j}| = \sqrt{3^2 + (-4)^2} = \sqrt{25} = 5$

The modulus bars mean length, computed exactly like coordinate distance. Keep surd answers exact: $|2\mathbf{i} + 3\mathbf{j}| = \sqrt{13}$, full stop, on Paper 1. The squared negative loses its sign inside the root, but write $(-4)^2$ with brackets, because $-4^2$ read carelessly is $-16$.

Unit vectors: direction distilled

A unit vector has magnitude $1$; the unit vector in $\mathbf{v}$‘s direction is $\dfrac{\mathbf{v}}{|\mathbf{v}|}$:

Unit vector along $3\mathbf{i} - 4\mathbf{j}$: $\dfrac{3\mathbf{i} - 4\mathbf{j}}{5} = 0.6\mathbf{i} - 0.8\mathbf{j}$

The construction the exam loves: speed × unit vector

“A particle moves with speed $26$ m/s in the direction of $5\mathbf{i} + 12\mathbf{j}$”, speed is a scalar; the velocity vector is built by pointing the speed along the unit vector:

$|5\mathbf{i} + 12\mathbf{j}| = 13$ $\to$ $\mathbf{v}$ $= 26 \times \dfrac{5\mathbf{i} + 12\mathbf{j}}{13} =$ $10\mathbf{i} + 24\mathbf{j}$

Three marks typically: the magnitude (M), the unit-vector step written out (M), the assembled velocity (A). Doing it in one mental hop risks all three. The reverse reading matters equally: given velocity $10\mathbf{i} + 24\mathbf{j}$, the speed is $|\mathbf{v}| = 26$, speed is the magnitude of velocity, a distinction kinematics questions exploit.

Same logic for “find a vector of magnitude $15$ parallel to $3\mathbf{i} - 4\mathbf{j}$”: $15 \times \dfrac{3\mathbf{i} - 4\mathbf{j}}{5} = 9\mathbf{i} - 12\mathbf{j}$, and note “parallel” admits the negative too ($-9\mathbf{i} + 12\mathbf{j}$) unless direction is specified; mention it.

Parallel vectors

$\mathbf{u}$ is parallel to $\mathbf{v}$ exactly when $\mathbf{u} = k\mathbf{v}$ for some scalar $k$. To test: are the components in equal ratio? This little fact is the engine of collinearity proofs later in the topic.

Common mistakes

• Magnitude computed without squaring the negative properly
• Unit-vector step skipped, gluing speed onto the unscaled direction vector
• Speed and velocity conflated
• “Parallel” answered with only one of the two directions
• Surds decimalised mid-question

Full topic context: Vectors notes.