Vector Algebra

The One Thing

Vector algebra represents quantities with both magnitude and direction and provides component-based methods for solving geometric problems. The dot product determines angles, projections, and perpendicularity; the cross product determines normals, areas, and orientation; and the scalar triple product determines volumes and coplanarity.

Definitions and Results

• Scalar: A quantity described completely by its magnitude, such as mass, time, temperature, or distance.
• Vector: A quantity having both magnitude and direction, such as displacement, velocity, force, or acceleration.
• Standard unit vectors: $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ point along the positive $x$-, $y$-, and $z$-axes, respectively, and satisfy $|\mathbf{i}|=|\mathbf{j}|=|\mathbf{k}|=1$.
• Magnitude of a vector: If $\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$, then

$$|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}.$$ The zero vector has magnitude zero and no definite direction.

• Unit vector: A vector of magnitude $1$ indicating direction. The unit vector in the direction of a non-zero vector $\mathbf{a}$ is

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

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

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

• Direction cosines: For $\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$, the cosines of its angles with the positive coordinate axes are

$$\frac{a_1}{|\mathbf{a}|},\frac{a_2}{|\mathbf{a}|},\frac{a_3}{|\mathbf{a}|}.$$

• Vector addition: Vectors are added component-wise:

$$(a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k})+(b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k})=(a_1+b_1)\mathbf{i}+(a_2+b_2)\mathbf{j}+(a_3+b_3)\mathbf{k}.$$ This corresponds geometrically to the triangle law and parallelogram law.

• Scalar multiplication: Multiplication by a scalar changes a vector’s magnitude and reverses its direction if the scalar is negative. For any scalar $\lambda$,

$$|\lambda \mathbf{a}|=|\lambda ||\mathbf{a}|.$$

• 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

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

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

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

• Vector equality: Two vectors are equal exactly when the corresponding coefficients of $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are equal.
• Scalar product or dot product: For vectors $\mathbf{a}$ and $\mathbf{b}$,

$$\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta =a_1{b}_1+a_2{b}_2+a_3{b}_3,$$ where $\theta$ is the angle between the non-zero vectors.

• Properties of the dot product: The dot product is commutative and distributive over vector addition, and

$$\mathbf{a}\cdot \mathbf{a}=|\mathbf{a}{|}^2.$$ Non-zero vectors are perpendicular if and only if $$\mathbf{a}\cdot \mathbf{b}=0.$$

• Projection: The scalar projection of $\mathbf{a}$ on $\mathbf{b}$, with $\mathbf{b}\ne 0$, is

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

• Vector product or cross product: The cross product $\mathbf{a}\times \mathbf{b}$ is perpendicular to both vectors, has magnitude

$$|\mathbf{a}\times \mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin \theta ,$$ and has direction given by the right-hand rule.

• Component formula for the cross product:

$$\mathbf{a}\times \mathbf{b}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$$ and therefore $$\mathbf{a}\times \mathbf{b}=(a_2{b}_3-a_3{b}_2)\mathbf{i}-(a_1{b}_3-a_3{b}_1)\mathbf{j}+(a_1{b}_2-a_2{b}_1)\mathbf{k}.$$

• Properties of the cross product: The cross product is anti-commutative:

$$\mathbf{a}\times \mathbf{b}=-(\mathbf{b}\times \mathbf{a}),$$ is distributive over addition, and satisfies $$\mathbf{a}\times \mathbf{a}=0.$$

• Area using the cross product: The area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$ is

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

• Collinear vectors: Two vectors are collinear if one is a scalar multiple of the other. For non-zero vectors, this is equivalent to

$$\mathbf{a}\times \mathbf{b}=0.$$ In particular, parallel vectors have zero cross product.

• Scalar triple product: The scalar triple product is

$$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c}).$$ Its absolute value is the volume of the parallelepiped formed by the three vectors. It can be calculated as a determinant and changes sign when any two vectors are interchanged.

• Coplanar vectors: Three vectors are coplanar if and only if

$$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=0.$$

• Volumes:

- The volume of a parallelepiped determined by $\mathbf{a},\mathbf{b},\mathbf{c}$ is $$|\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})|.$$ - The volume of a tetrahedron determined by vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ from a common vertex is $$\frac{1}{6}|\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})|.$$

Worked Methods

1. Expressing vectors in component form

• Write a vector using its coefficients of $\mathbf{i},\mathbf{j},\mathbf{k}$:

$$\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}.$$

• Treat the coefficients as the vector’s $x$-, $y$-, and $z$-components.
• Use equality of corresponding coefficients when comparing vectors.

For a point $P(x,y,z)$, its position vector is $$\overrightarrow{OP}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}.$$

2. Finding magnitude, unit vector, and direction cosines

For $$\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}:$$

• Calculate the magnitude:

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

• If $\mathbf{a}\ne 0$, divide by its magnitude to obtain the unit vector:

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

• Obtain the direction cosines from

$$\frac{a_1}{|\mathbf{a}|},\frac{a_2}{|\mathbf{a}|},\frac{a_3}{|\mathbf{a}|}.$$

3. Adding, subtracting, and scaling vectors

For $$\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k},\mathbf{b}=b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}:$$

• Add or subtract corresponding components:

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

• For subtraction, use

$$\mathbf{a}-\mathbf{b}=(a_1-b_1)\mathbf{i}+(a_2-b_2)\mathbf{j}+(a_3-b_3)\mathbf{k}.$$

• For a scalar $\lambda$, multiply every component:

$$\lambda \mathbf{a}=\lambda a_1\mathbf{i}+\lambda a_2\mathbf{j}+\lambda a_3\mathbf{k}.$$

• Interpret a negative scalar as a reversal of direction.

The distance between the corresponding points is then $$|\mathbf{a}-\mathbf{b}|.$$

4. Using the section formula

If a point divides the line joining position vectors $\mathbf{a}$ and $\mathbf{b}$ internally in the ratio $m:n$:

• Identify the ratio $m:n$.
• Substitute into

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

• Simplify the coefficients of $\mathbf{i},\mathbf{j},\mathbf{k}$.

5. Finding an angle or testing perpendicularity with the dot product

For non-zero vectors $\mathbf{a}$ and $\mathbf{b}$:

• Calculate the component dot product:

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

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

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

• Determine $\theta$ if required.
• For perpendicularity, test whether

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

6. Finding projections

To project $\mathbf{a}$ onto $\mathbf{b}$, with $\mathbf{b}\ne 0$:

• Calculate $\mathbf{a}\cdot \mathbf{b}$.
• Calculate $|\mathbf{b}|$ or $|\mathbf{b}{|}^2$.
• Use the scalar projection formula:

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

• Use the vector projection formula when a vector is required:

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

7. Computing a cross product

For $$\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k},\mathbf{b}=b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}:$$

• Set up 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 it as

$$(a_2{b}_3-a_3{b}_2)\mathbf{i}-(a_1{b}_3-a_3{b}_1)\mathbf{j}+(a_1{b}_2-a_2{b}_1)\mathbf{k}.$$

• Interpret the result as a vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.
• Use the right-hand rule to determine its orientation if required.

8. Finding areas with a cross product

For a parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$:

• Calculate $\mathbf{a}\times \mathbf{b}$.
• Find its magnitude.
• Use

$$\text{area of parallelogram}=|\mathbf{a}\times \mathbf{b}|.$$

For the corresponding triangle:

• Use the same cross product.
• Multiply its magnitude by $\frac{1}{2}$:

$$\text{area of triangle}=\frac{1}{2}|\mathbf{a}\times \mathbf{b}|.$$

For a triangle with vertices having position vectors $\mathbf{a},\mathbf{b},\mathbf{c}$:

• Form the side vectors

$$\mathbf{b}-\mathbf{a},\mathbf{c}-\mathbf{a}.$$

• Compute

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

• Use

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

9. Testing collinearity

• Form the relevant vectors.
• Determine whether one vector is a scalar multiple of the other.
• Alternatively, for non-zero vectors, calculate the cross product.
• Conclude collinearity if

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

10. Finding volumes and testing coplanarity

For vectors $\mathbf{a},\mathbf{b},\mathbf{c}$:

• Calculate $\mathbf{b}\times \mathbf{c}$.
• Take the dot product with $\mathbf{a}$:

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

• Take the absolute value for the parallelepiped volume:

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

• For a tetrahedron from a common vertex, divide by $6$:

$$V_{\text{tetrahedron}}=\frac{1}{6}|\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})|.$$

• Test coplanarity by checking whether

$$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=0.$$

Where It Goes Wrong

• Confusing scalars and vectors: mass, time, temperature, and distance have magnitude only, whereas displacement, velocity, force, and acceleration also have direction.
• Omitting the non-zero condition when forming the unit vector $\mathbf{a}/|\mathbf{a}|$ or using the angle formula $\cos \theta =(\mathbf{a}\cdot \mathbf{b})/(|\mathbf{a}||\mathbf{b}|)$.
• Forgetting that scalar multiplication by a negative scalar reverses direction, while always satisfying $|\lambda \mathbf{a}|=|\lambda ||\mathbf{a}|$.
• Using the dot product where the cross product is required: perpendicularity uses $\mathbf{a}\cdot \mathbf{b}=0$, whereas collinearity and area calculations use $\mathbf{a}\times \mathbf{b}$.
• Losing the negative sign in the $\mathbf{j}$-component when expanding

$$\mathbf{a}\times \mathbf{b}=(a_2{b}_3-a_3{b}_2)\mathbf{i}-(a_1{b}_3-a_3{b}_1)\mathbf{j}+(a_1{b}_2-a_2{b}_1)\mathbf{k}.$$

• Omitting absolute values in area and volume formulas, or omitting the factor $\frac{1}{2}$ for a triangle and $\frac{1}{6}$ for a tetrahedron.
• Treating the scalar triple product as unchanged under interchange of two vectors; it changes sign, although its absolute value gives the volume.

What Gets Asked

• Define and distinguish scalars, vectors, magnitudes, unit vectors, position vectors, and direction cosines.
• Express points and vectors in $\mathbf{i},\mathbf{j},\mathbf{k}$ component form.
• Add, subtract, and multiply vectors by scalars.
• Find the distance between points from their position vectors.
• Apply the internal section formula to find a point dividing a line in a given ratio.
• Calculate magnitudes, unit vectors, and direction cosines.
• Find the angle between two non-zero vectors using the dot product.
• Test whether vectors are perpendicular using $\mathbf{a}\cdot \mathbf{b}=0$.
• Find scalar and vector projections.
• Calculate a cross product using the determinant formula.
• Determine a vector perpendicular to two given vectors and its orientation using the right-hand rule.
• Find parallelogram and triangle areas using cross products.
• Find the area of a triangle with position vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ using

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

• Test whether two vectors are collinear.
• Calculate parallelepiped and tetrahedron volumes using the scalar triple product.
• Test whether three vectors are coplanar.
• Use vector equalities by equating the corresponding coefficients of $\mathbf{i},\mathbf{j},\mathbf{k}$.