IB Math: Vectors Fundamentals

Vectors are the language of direction and magnitude, essential for describing everything from forces in physics to navigation in computer graphics. In the IB Mathematics curriculum, mastering vector fundamentals unlocks your ability to solve complex geometric and physical problems in two and three dimensions, providing a powerful toolkit for both analysis and application.

Vector Notation and Basic Arithmetic

A vector is a quantity possessing both magnitude (size) and direction. This distinguishes it from a scalar, which has only magnitude. In print, vectors are often denoted by bold lowercase letters (e.g., $\mathbf{a}$, $\mathbf{v}$) or with an overhead arrow (e.g., $\overrightarrow{AB}$). The vector from point $A$ to point $B$ is written as $\overrightarrow{AB}$. In component form, a vector in three dimensions is written as a column: $\mathbf{a}=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$ or as an ordered triple $\mathbf{a}=(x,y,z)$. The position vector of a point $P$ is the vector $\overrightarrow{OP}$ from the origin $O$ to $P$.

Vector arithmetic follows specific geometric rules. Vector addition is done tip-to-tail: placing the tail of the second vector at the tip of the first; the resultant vector is from the tail of the first to the tip of the second. Algebraically, you add corresponding components: $$\mathbf{a}+\mathbf{b}=\left(\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right)+\left(\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right)=\left(\begin{array}{c}{a}_1+b_1\\ a_2+b_2\\ a_3+b_3\end{array}\right).$$ Vector subtraction $\mathbf{a}-\mathbf{b}$ is equivalent to $\mathbf{a}+(-\mathbf{b})$, where $-\mathbf{b}$ is the vector with the same magnitude as $\mathbf{b}$ but opposite direction. Scalar multiplication involves multiplying each component by a scalar (a real number) $k$: $k\mathbf{a}=\left(\begin{array}{c}k{a}_1\\ k{a}_2\\ k{a}_3\end{array}\right)$. This scales the vector's length by a factor of $|k|$ and reverses its direction if $k$ is negative.

The Scalar (Dot) Product and Angles

The scalar product (or dot product) is a fundamental operation that yields a scalar from two vectors. It is defined as $\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$. Crucially, it also has a geometric definition: $\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta$, where $\theta$ is the angle between the two vectors and $|\mathbf{a}|$ represents the magnitude (or length) of $\mathbf{a}$, calculated as $|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}$.

These two definitions are interchangeable and provide a powerful method for finding the angle between any two vectors. You rearrange the geometric formula to solve for $\cos \theta$: $$\cos \theta =\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}.$$ For example, to find the angle between $\mathbf{u}=(2,-1,3)$ and $\mathbf{v}=(4,0,-2)$, first compute the scalar product: $\mathbf{u}\cdot \mathbf{v}=(2)(4)+(-1)(0)+(3)(-2)=8+0-6=2$. Then find the magnitudes: $|\mathbf{u}|=\sqrt{4+1+9}=\sqrt{14}$ and $|\mathbf{v}|=\sqrt{16+0+4}=\sqrt{20}$. Finally, $\cos \theta =\frac{2}{\sqrt{14}\sqrt{20}}=\frac{2}{\sqrt{280}}$, so $\theta =\arccos \left(\frac{2}{\sqrt{280}}\right)$.

Geometric Applications in 2D and 3D

Vectors provide an elegant framework for solving geometric problems. A core application is proving parallelism and perpendicularity. Two non-zero vectors are parallel if one is a scalar multiple of the other: $\mathbf{a}=k\mathbf{b}$. They are perpendicular (or orthogonal) if their scalar product is zero: $\mathbf{a}\cdot \mathbf{b}=0$. This follows directly from the geometric definition, as $\cos 90^{\circ }=0$.

Finding midpoints is straightforward with vectors. The midpoint $M$ of a line segment with endpoints $A$ and $B$ (with position vectors $\mathbf{a}$ and $\mathbf{b}$) has a position vector given by the average: $$\overrightarrow{OM}=\frac{1}{2}(\mathbf{a}+\mathbf{b}).$$ This formula extends easily to finding a point that divides a segment in any given ratio.

You can also use vectors to find the distance between points, which is simply the magnitude of the vector connecting them: $AB=|\overrightarrow{AB}|=|\mathbf{b}-\mathbf{a}|$. More advanced applications include finding the foot of a perpendicular from a point to a line or checking if three points are collinear (by seeing if the vectors between them are parallel).

Common Pitfalls

1. Confusing Points and Vectors: A point is a location; a vector is a displacement. The coordinates $(3,4)$ describe a point. The components $\left(\begin{array}{c}3\\ 4\end{array}\right)$ describe a vector. The vector $\overrightarrow{AB}$ is found by subtracting coordinates: $\mathbf{b}-\mathbf{a}$. A common mistake is to add them instead.
2. Mishandling the Scalar Product: Remember the result of $\mathbf{a}\cdot \mathbf{b}$ is a scalar, not a vector. You cannot take a dot product of a scalar and a vector. Furthermore, when using the formula $\cos \theta =\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$, you must ensure you are not accidentally using the wrong vectors, such as using a position vector when a direction vector is required.
3. Dimensional Inconsistency: A vector in 3D has three components. You cannot take the scalar product of a 2D vector and a 3D vector directly. In geometric problems, all vectors involved in an operation (addition, subtraction, dot product) must be of the same dimension. Always check that your vectors "live" in the same space.
4. Misinterpreting Parallelism: Stating vectors are parallel because their components have the same sign is incorrect. They are parallel only if the ratios of their corresponding components are equal. For $\mathbf{u}=(2,4,6)$ and $\mathbf{v}=(1,2,3)$, the ratios are $2/1=4/2=6/3=2$, so they are parallel ($\mathbf{u}=2\mathbf{v}$). For $\mathbf{w}=(2,4,5)$, the ratio for the z-component is $6/5$, which differs, so they are not parallel.

Summary

• Vectors are defined by magnitude and direction, and are manipulated through addition (tip-to-tail), subtraction, and scalar multiplication, all performed component-wise.
• The scalar product $\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3=|\mathbf{a}||\mathbf{b}|\cos \theta$ is used to calculate the angle between vectors and test for perpendicularity.
• Two vectors are parallel if one is a scalar multiple of the other, and perpendicular if their scalar product equals zero.
• Vector methods provide efficient solutions to geometric problems, including finding midpoints $\left(\frac{1}{2}(\mathbf{a}+\mathbf{b})\right)$, distances, and proving collinearity or specific properties of shapes.
• Always maintain a clear distinction between points and vectors, ensure dimensional consistency in operations, and apply the correct vector definitions to avoid common errors in calculation and interpretation.