- Chapter 10
- Vectors and Three-Dimensional Geometry
- 23 questions
Vector Algebra
Vectors and scalars, direction cosines and ratios, types of vectors, addition, section formula, and the scalar and vector products.
Some quantities need a direction to be meaningful — a displacement, a force. A vector carries both magnitude and direction, and vector algebra is the arithmetic that results.
Two products dominate the chapter and they answer different questions. The scalar product returns a number and measures how much two vectors point the same way, which is what makes it the tool for angles and projections. The vector product returns a vector perpendicular to both, which is what makes it the tool for areas and for anything requiring a normal direction.
What you should be able to do
- Distinguish vectors from scalars and identify equal, unit, zero, parallel and collinear vectors.
- Find the magnitude, direction cosines and direction ratios of a vector.
- Add vectors and multiply by a scalar, and use the section formula.
- Compute the scalar product, and use it for angles and projections.
- Compute the vector product, and use it for areas and perpendicular directions.
Topics
5 topics
- Vectors and Scalars
Magnitude and direction, and the standard types of vector including position vectors and components.
- Types of vectors
- Position vector and components
- Direction Cosines and Direction Ratios
Defining the direction of a vector numerically, and the relation.
- Addition of Vectors and the Section Formula
Triangle and parallelogram laws, scalar multiplication, and the position vector of a dividing point.
- Scalar (Dot) Product
Definition, geometric meaning, properties, and applications to angles and projections.
- Vector (Cross) Product
Definition, properties, and applications to area and to finding a perpendicular direction.
Formulas
13 items · All formulas
Magnitude of a vector
\left|\vec{a}\right| = \sqrt{a_1^{2} + a_2^{2} + a_3^{2}} \quad \text{for } \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}Unit vector
\hat{a} = \dfrac{\vec{a}}{\left|\vec{a}\right|}- Valid when
- \vec{a} \neq \vec{0}.
Scalar (dot) product
\vec{a}\cdot\vec{b} = \left|\vec{a}\right|\left|\vec{b}\right|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3Angle between two vectors
\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{\left|\vec{a}\right|\left|\vec{b}\right|}Perpendicularity test
\vec{a} \perp \vec{b} \iff \vec{a}\cdot\vec{b} = 0- Valid when
- \vec{a}, \vec{b} non-zero.
Projection of one vector on another
\text{Projection of } \vec{a} \text{ on } \vec{b} = \dfrac{\vec{a}\cdot\vec{b}}{\left|\vec{b}\right|}A signed scalar — negative when the angle is obtuse.
Vector (cross) product
\vec{a}\times\vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}Magnitude of the cross product
\left|\vec{a}\times\vec{b}\right| = \left|\vec{a}\right|\left|\vec{b}\right|\sin\thetaAnti-commutativity
\vec{a}\times\vec{b} = -\left(\vec{b}\times\vec{a}\right)Order reverses direction — unlike the dot product, which commutes.
Area of a triangle from two sides
\text{Area} = \dfrac{1}{2}\left|\vec{a}\times\vec{b}\right|Area of a parallelogram
\text{Area} = \left|\vec{a}\times\vec{b}\right|With diagonals \vec{d_1}, \vec{d_2}, the area is \frac{1}{2}\left|\vec{d_1}\times\vec{d_2}\right|.
Direction cosines
l = \dfrac{a_1}{|\vec{a}|}, \quad m = \dfrac{a_2}{|\vec{a}|}, \quad n = \dfrac{a_3}{|\vec{a}|}, \qquad l^{2}+m^{2}+n^{2} = 1Section formula (internal division)
\vec{r} = \dfrac{m\vec{b} + n\vec{a}}{m + n}Position vector of the point dividing AB internally in the ratio m:n.
Key points
6 items · All key points
The dot product gives a scalar; the cross product gives a vector. Check your answer is the right kind of object.
\vec{a}\cdot\vec{b} = 0 means perpendicular; \vec{a}\times\vec{b} = \vec{0} means parallel.
Both require the vectors to be non-zero for the conclusion to follow.
The cross product is anti-commutative: reversing the order reverses the direction.
The area of a triangle carries a factor \frac{1}{2}; the area of a parallelogram does not.
A projection is a signed scalar and may be negative — do not take its modulus unless asked.
Direction cosines always satisfy l^{2}+m^{2}+n^{2}=1; direction ratios need not.
Use the identity as a check on any set of direction cosines you compute.
Common mistakes
5 items · All common mistakes
- Mistake
Giving a vector as the answer to a dot product, or a scalar as the answer to a cross product.
Instead\vec{a}\cdot\vec{b} is a number; \vec{a}\times\vec{b} is a vector. Check the type before writing the final line.
WhyBoth are called 'products' and are computed from the same components.
- Mistake
Writing \vec{a}\times\vec{b} = \vec{b}\times\vec{a}.
Instead\vec{a}\times\vec{b} = -\left(\vec{b}\times\vec{a}\right).
WhyCommutativity holds for the dot product, and the habit carries over.
- Mistake
Omitting the factor \frac{1}{2} when finding the area of a triangle.
InsteadTriangle: \frac{1}{2}\left|\vec{a}\times\vec{b}\right|. Parallelogram: \left|\vec{a}\times\vec{b}\right|.
- Mistake
Making a sign error in the middle term when expanding the cross-product determinant.
InsteadThe \hat{j} component carries a minus sign: \vec{a}\times\vec{b} = (a_2b_3-a_3b_2)\hat{i} - (a_1b_3-a_3b_1)\hat{j} + (a_1b_2-a_2b_1)\hat{k}.
WhyThe alternating sign pattern of cofactor expansion is easy to lose in a hurry.
- Mistake
Confusing direction ratios with direction cosines.
InsteadDirection cosines are the components of the unit vector; divide the direction ratios by the magnitude.