How Do You Write a Vector Using Base Vectors i, j, k?

A vector with components $a$, $b$, and $c$ is written using base vectors as $\mathbf{v}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$. The coefficients represent movement in the $x$-, $y$-, and $z$-directions respectively.

This is IB Mathematics: Analysis and Approaches HL content from AHL 3.12: Vector definitions. It can be assessed on Paper 1, Paper 2, or HL Paper 3.

The base vectors are unit vectors, so each has magnitude $1$ and points along one positive coordinate axis:

Because any vector is the sum of its independent coordinate movements, its component form and base-vector form are equivalent:

$$(a,b,c)=a(1,0,0)+b(0,1,0)+c(0,0,1)=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}.$$

For example, $(3,-2,5)$ becomes $\mathbf{v}=3\mathbf{i}-2\mathbf{j}+5\mathbf{k}$. This means moving $3$ units in the positive -direction, units in the negative -direction, and units in the positive -direction. A negative component reverses direction; it does not make the vector invalid.

The same representation also makes vector arithmetic straightforward. Add or subtract corresponding coefficients, and multiply every coefficient when using a scalar. The magnitude is $\lvert\mathbf{v}\rvert=\sqrt{a^2+b^2+c^2}$, because the three base vectors are mutually perpendicular unit vectors. For , . Keeping each coordinate in its fixed position prevents accidental interchange of the - and -components during longer calculations and keeps exam working clear.

To find a displacement vector from $A(1,4,-2)$ to $B(5,1,3)$, subtract the position vector of $A$ from that of $B$:

$$\overrightarrow{AB}=(5-1,1-4,3-(-2))=(4,-3,5)=4\mathbf{i}-3\mathbf{j}+5\mathbf{k}.$$

The order matters: $\overrightarrow{AB}=B-A$, whereas $\overrightarrow{BA}=A-B=-\overrightarrow{AB}$. Reversing the subtraction therefore produces a vector with the same magnitude but the opposite direction.

A common misconception is that a point and a vector are identical. A point specifies a location, while a vector specifies magnitude and direction, although both can be represented using three coordinates.

Exam technique: If asked to “write” or “express” a vector in terms of $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, include every coefficient with its correct sign. For displacement, write the subtraction step before simplifying; this demonstrates the required method and reduces sign errors.