Adding & Scaling Vectors

Adding, subtracting and scaling vectors are the building blocks behind resultants, position vectors and the parallel condition. All three work component by component.

Adding and subtracting

Add (or subtract) the $\mathbf{i}$-components together and the $\mathbf{j}$-components together, separately: $(3\mathbf{i} + 2\mathbf{j}) + (\mathbf{i} - 5\mathbf{j}) = (3 + 1)\mathbf{i} + (2 - 5)\mathbf{j} = 4\mathbf{i} - 3\mathbf{j}$

The sum is the resultant. Subtraction is the same with the signs: $\mathbf{a} - \mathbf{b}$ adds the negative of $\mathbf{b}$.

Scaling (scalar multiplication)

Multiplying a vector by a scalar multiplies both components: $3(2\mathbf{i} - \mathbf{j}) = 6\mathbf{i} - 3\mathbf{j}$

Scaling changes a vector’s length but not its direction (a negative scalar reverses the direction). This is exactly why two vectors are parallel when one is a scalar multiple of the other.

Combining the two

Most vector work combines addition and scaling, for example forming $\lambda\mathbf{a} + \mu\mathbf{b}$: $2(1\mathbf{i} + 1\mathbf{j}) + 3(1\mathbf{i} - 1\mathbf{j}) = (2 + 3)\mathbf{i} + (2 - 3)\mathbf{j} = 5\mathbf{i} - \mathbf{j}$

Expressing one vector as such a combination of two others is done by comparing components, which relies on exactly this arithmetic.

Keep the signs straight

The single most common error is a sign slip on a component: $2 + (-5) = -3$, not $+3$ or $-7$. Since both a resultant and its magnitude depend on the components, one wrong sign corrupts both.

Common mistakes

• Adding an $\mathbf{i}$-component to a $\mathbf{j}$-component.
• Scaling only one component instead of both.
• Sign errors when subtracting or with a negative scalar.

Full topic context: Vectors notes, and Vector Notation & Magnitude.