Vectors

The One Thing

Vectors encode both magnitude and direction and can be manipulated algebraically through components. The dot product gives scalar information about angles and projections, whereas the cross product gives a perpendicular vector whose magnitude represents an area.

Definitions and Results

• Scalar: A quantity described only by magnitude, such as mass, time, temperature, or speed.
• Vector: A quantity described by both magnitude and direction, such as displacement, velocity, acceleration, or force.
• Magnitude or Modulus: For $\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$,

$$|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}.$$

• Unit Vector: A vector of magnitude $1$. The unit vector in the direction of a nonzero vector $\mathbf{a}$ is

$$\frac{\mathbf{a}}{|\mathbf{a}|}.$$

• Position Vector: The vector from the origin to a point. For $P(x,y,z)$,

$$\overrightarrow{OP}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}.$$

• Equal Vectors: Vectors with the same magnitude and direction, even when located at different positions.
• Negative Vector: The vector $-\mathbf{a}$ has the same magnitude as $\mathbf{a}$, but the opposite direction.
• Collinear or Parallel Vectors: Nonzero vectors $\mathbf{a}$ and $\mathbf{b}$ are parallel when

$$\mathbf{a}=\lambda \mathbf{b}$$ for some scalar $\lambda$.

• Vector Addition:

$$(a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k})+(b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k})=(a_1+b_1)\mathbf{i}+(a_2+b_2)\mathbf{j}+(a_3+b_3)\mathbf{k}.$$ It is commutative and associative: $$\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a},(\mathbf{a}+\mathbf{b})+\mathbf{c}=\mathbf{a}+(\mathbf{b}+\mathbf{c}).$$

• Scalar Multiplication: Multiplication by a scalar changes a vector’s magnitude and reverses its direction if the scalar is negative. It satisfies

$$\lambda (\mathbf{a}+\mathbf{b})=\lambda \mathbf{a}+\lambda \mathbf{b},(\lambda +\mu )\mathbf{a}=\lambda \mathbf{a}+\mu \mathbf{a}.$$

• Section Formula: If a point divides the line joining position vectors $\mathbf{a}$ and $\mathbf{b}$ internally in the ratio $m:n$, its position vector is

$$\frac{n\mathbf{a}+m\mathbf{b}}{m+n}.$$

• Distance Between Position Vectors: The distance between points with position vectors $\mathbf{a}$ and $\mathbf{b}$ is

$$|\mathbf{a}-\mathbf{b}|.$$

• Dot Product or Scalar Product:

$$\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta ,$$ where $\theta$ is the angle between the vectors. The result is a scalar.

• Component Form of the Dot Product:

$$(a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k})\cdot (b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k})=a_1{b}_1+a_2{b}_2+a_3{b}_3.$$

• Dot-Product Properties:

$$\mathbf{a}\cdot \mathbf{b}=\mathbf{b}\cdot \mathbf{a},$$ $$\mathbf{a}\cdot (\mathbf{b}+\mathbf{c})=\mathbf{a}\cdot \mathbf{b}+\mathbf{a}\cdot \mathbf{c}.$$ For coordinate unit vectors, $$\mathbf{i}\cdot \mathbf{i}=\mathbf{j}\cdot \mathbf{j}=\mathbf{k}\cdot \mathbf{k}=1,$$ and $$\mathbf{i}\cdot \mathbf{j}=\mathbf{j}\cdot \mathbf{k}=\mathbf{k}\cdot \mathbf{i}=0.$$

• Angle Between Nonzero Vectors:

$$\cos \theta =\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}.$$ If $\mathbf{a}\cdot \mathbf{b}>0$, the angle is acute; if $\mathbf{a}\cdot \mathbf{b}=0$, the vectors are perpendicular; if $\mathbf{a}\cdot \mathbf{b}<0$, the angle is obtuse.

• Perpendicular Vectors: Nonzero vectors are perpendicular when

$$\mathbf{a}\cdot \mathbf{b}=0.$$

• Projection: The scalar projection of $\mathbf{a}$ on $\mathbf{b}$ is

$$\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{b}|},$$ and the vector projection is $$\left(\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{b}{|}^2}\right)\mathbf{b}.$$

• Cross Product or Vector Product: For vectors in three-dimensional space,

$$|\mathbf{a}\times \mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin \theta ,$$ and $\mathbf{a}\times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.

• Direction of the Cross Product: The direction of $\mathbf{a}\times \mathbf{b}$ is determined by the right-hand rule and is perpendicular to the plane containing $\mathbf{a}$ and $\mathbf{b}$.
• Cross-Product Properties:

$$\mathbf{a}\times \mathbf{b}=-(\mathbf{b}\times \mathbf{a}),$$ and the cross product is distributive over vector addition. Also, $$\mathbf{a}\times \mathbf{a}=0.$$

• Coordinate-Unit-Vector Cross Products:

$$\mathbf{i}\times \mathbf{j}=\mathbf{k},\mathbf{j}\times \mathbf{k}=\mathbf{i},\mathbf{k}\times \mathbf{i}=\mathbf{j}.$$ Reversing the order changes the sign.

• Parallelism Criterion: Two nonzero vectors are parallel if and only if

$$\mathbf{a}\times \mathbf{b}=0.$$

• Area Using the Cross Product: The parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$ has area

$$|\mathbf{a}\times \mathbf{b}|,$$ while the triangle formed by them has area $$\frac{1}{2}|\mathbf{a}\times \mathbf{b}|.$$

• Triangle Area from Position Vectors: For vertices with position vectors $\mathbf{a},\mathbf{b},\mathbf{c}$, the area is

$$\frac{1}{2}|(\mathbf{b}-\mathbf{a})\times (\mathbf{c}-\mathbf{a})|.$$

• Scalar Triple Product: The expression

$$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$$ is a scalar whose absolute value is the volume of the parallelepiped formed by the three vectors.

• Determinant Form of the Scalar Triple Product:

$$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|.$$ It is zero if and only if the three vectors are coplanar. Interchanging any two vectors changes its sign, but not its absolute value.

• Vector Triple Product:

$$\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\mathbf{b}(\mathbf{a}\cdot \mathbf{c})-\mathbf{c}(\mathbf{a}\cdot \mathbf{b}).$$

Worked Methods

1. Representing and Finding the Magnitude of a Vector

• Write a three-dimensional vector in Cartesian form:

$$\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k},$$ where $\mathbf{i},\mathbf{j},\mathbf{k}$ are unit vectors along the positive $x$-, $y$-, and $z$-axes.

• Calculate its magnitude using

$$|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}.$$

• If a unit vector is required, divide by the magnitude:

$$\frac{\mathbf{a}}{|\mathbf{a}|}.$$

2. Adding, Subtracting, and Scaling Vectors

• Add or subtract corresponding components.
• For scalar multiplication, multiply every component by the scalar.
• Interpret a negative scalar as a reversal of direction.
• For points with position vectors $\mathbf{a}$ and $\mathbf{b}$, obtain the connecting displacement vector from

$$\mathbf{b}-\mathbf{a},$$ and its distance from $$|\mathbf{b}-\mathbf{a}|.$$

3. Applying the Section Formula

• Identify the endpoint position vectors $\mathbf{a}$ and $\mathbf{b}$.
• Identify the internal division ratio $m:n$.
• Substitute into

$$\frac{n\mathbf{a}+m\mathbf{b}}{m+n}.$$

• Simplify the resulting vector component by component.

4. Finding an Angle Using the Dot Product

• Calculate the dot product using either

$$\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta$$ or $$\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3.$$

• Calculate $|\mathbf{a}|$ and $|\mathbf{b}|$.
• Use

$$\cos \theta =\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}.$$

• Determine $\theta$, taking account of the sign of the dot product: acute for a positive value, right-angled for zero, and obtuse for a negative value.

5. Testing Perpendicularity

• Calculate $\mathbf{a}\cdot \mathbf{b}$.
• Conclude that the nonzero vectors are perpendicular if and only if

$$\mathbf{a}\cdot \mathbf{b}=0.$$

6. Finding a Projection

• Calculate $\mathbf{a}\cdot \mathbf{b}$.
• Calculate $|\mathbf{b}|$ or $|\mathbf{b}{|}^2$.
• For the scalar projection of $\mathbf{a}$ on $\mathbf{b}$, use

$$\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{b}|}.$$

• For the vector projection, use

$$\left(\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{b}{|}^2}\right)\mathbf{b}.$$

7. Calculating a Cross Product

• Confirm that the vectors are in three-dimensional space.
• Use the determinant

$$\mathbf{a}\times \mathbf{b}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|.$$

• Expand carefully, remembering the sign associated with the $\mathbf{j}$-component.
• Interpret the resulting vector as perpendicular to both original vectors.
• Use the right-hand rule to check its direction.
• Remember that reversing the order gives

$$\mathbf{b}\times \mathbf{a}=-(\mathbf{a}\times \mathbf{b}).$$

8. Finding Areas with the Cross Product

• Use the two vectors forming the parallelogram.
• Calculate their cross product.
• Take its magnitude:

$$|\mathbf{a}\times \mathbf{b}|.$$

• For a triangle, halve the result:

$$\frac{1}{2}|\mathbf{a}\times \mathbf{b}|.$$

• For a triangle with position vectors $\mathbf{a},\mathbf{b},\mathbf{c}$, form

$$(\mathbf{b}-\mathbf{a})\times (\mathbf{c}-\mathbf{a})$$ before taking half its magnitude.

9. Calculating a Scalar Triple Product and Testing Coplanarity

• Arrange the three vectors as rows in a determinant:

$$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|.$$

• Evaluate the determinant.
• Take the absolute value if the volume of the parallelepiped is required.
• If the result is zero, conclude that the three vectors are coplanar.
• If two vectors are interchanged, expect the scalar triple product to change sign, although the volume remains unchanged.

10. Simplifying a Vector Triple Product

• Recognise the expression as

$$\mathbf{a}\times (\mathbf{b}\times \mathbf{c}).$$

• Apply

$$\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\mathbf{b}(\mathbf{a}\cdot \mathbf{c})-\mathbf{c}(\mathbf{a}\cdot \mathbf{b}).$$

• Evaluate the two dot products.
• Form the indicated linear combination of $\mathbf{b}$ and $\mathbf{c}$.

Where It Goes Wrong

• Treating a scalar, such as speed, as a vector; vectors require both magnitude and direction.
• Using $\sqrt{a_1^2+a_2^2+a_3^2}$ without first identifying the vector components in the form $a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$.
• Forgetting that the dot product produces a scalar, whereas the cross product produces a vector perpendicular to both original vectors.
• Omitting the condition that perpendicularity and parallelism criteria apply to nonzero vectors.
• Reversing the order in a cross product without changing the sign; $\mathbf{a}\times \mathbf{b}=-(\mathbf{b}\times \mathbf{a})$.
• Using $|\mathbf{a}\times \mathbf{b}|$ directly for a triangle area instead of taking one half, or forgetting that the scalar triple product must be taken in absolute value for volume.

What Gets Asked

• Define scalar, vector, magnitude, unit vector, position vector, equal vector, negative vector, and parallel vector.
• Express a vector in Cartesian component form and calculate its magnitude or associated unit vector.
• Add, subtract, and scale vectors, including finding distances between position vectors.
• Apply the section formula to find a position vector dividing a line internally in a specified ratio.
• Calculate dot products from components and use them to find angles.
• Determine whether vectors are perpendicular and classify the angle as acute, right, or obtuse from the sign of the dot product.
• Find scalar and vector projections.
• Calculate a cross product using the determinant form and determine its direction using the right-hand rule.
• Test whether two nonzero vectors are parallel using their cross product.
• Calculate parallelogram and triangle areas using a cross product.
• Find the area of a triangle from three position vectors.
• Evaluate scalar triple products using determinants, calculate volumes, and test whether vectors are coplanar.
• Apply the vector triple product identity

$$\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\mathbf{b}(\mathbf{a}\cdot \mathbf{c})-\mathbf{c}(\mathbf{a}\cdot \mathbf{b}).$$