Precalculus Formula Sheet - Vectors (Chapter 11)
Vectors are mathematical objects that have both magnitude (size) and direction. They're essential in physics, engineering, and many other fields. This formula sheet covers the key concepts and formulas related to vectors that you'll encounter in Precalculus Chapter 11.
Vector Basics
A vector can be represented in various ways:
- Geometrically: an arrow with a specific length and direction
- Notation: v or v
- Component form: v = v, v in 2D, v = v, v, v in 3D
- Unit vector notation: v = xi + yj in 2D, v = xi + yj + zk in 3D
For a 2D vector v = v, v:
v = vi + vj
Where i = 1, 0 and j = 0, 1
Vector Magnitude
The magnitude (or length) of a vector is a scalar value that indicates its size:
For a 2D vector v = v, v:
|v| = (v + v)
For a 3D vector v = v, v, v:
|v| = (v + v + v)
Example: Find the magnitude of v = 3, 4
Solution: |v| = (3 + 4) = (9 + 16) = 25 = 5
Vector Direction
The direction of a vector can be expressed in different ways:
For a 2D vector v = v, v, the direction angle (from the positive x-axis) is:
tan() = v/v
Therefore: = arctan(v/v), adjusting for the appropriate quadrant
Example: Find the direction angle of v = 1, 3
Solution: tan() = 3/1 = 3, which means = 60
Unit Vectors
A unit vector has a magnitude of 1 and points in a specific direction:
The unit vector in the direction of a given vector v:
u = v/|v|
Note: The standard unit vectors in 3D space are i = 1, 0, 0, j = 0, 1, 0, and k = 0, 0, 1
Vector Addition and Subtraction
Vectors can be added or subtracted component-wise:
For vectors v = v, v and w = w, w:
v + w = v + w, v + w
v - w = v - w, v - w
Example: If v = 2, 3 and w = 5, 1, find v + w
Solution: v + w = 2 + 5, 3 + 1 = 7, 4
Scalar Multiplication
Multiplying a vector by a scalar (a real number) changes its magnitude but not its direction (unless the scalar is negative):
For a scalar c and a vector v = v, v:
cv = cv, cv
Dot Product
The dot product (or scalar product) of two vectors is a scalar value:
For vectors v = v, v and w = w, w:
v w = vw + vw
Geometric definition:
v w = |v| |w| cos()
Where is the angle between the vectors
Example: Find the dot product of v = 2, 3 and w = 5, 1
Solution: v w = 2(5) + 3(1) = 10 + 3 = 13
Important properties:
- v v = |v|
- v w = 0 if and only if v is orthogonal (perpendicular) to w
- v w = w v (Commutative property)
Finding the Angle Between Vectors
Using the dot product, we can find the angle between two vectors:
cos() = (v w)/(|v| |w|)
Therefore: = arccos[(v w)/(|v| |w|)]
Example: Find the angle between v = 1, 0 and w = 1, 3
Solution: |v| = 1, |w| = (1 + 3) = 2, v w = 1(1) + 0(3) = 1
cos() = 1/(1 2) = 0.5
= arccos(0.5) = 60
Vector Projection
The projection of vector v onto vector w (also called the component of v along w):
projwv = ((v w)/|w|) w
The scalar component of v along w is:
compwv = (v w)/|w|
Cross Product (3D Vectors Only)
The cross product of two 3D vectors results in a third vector orthogonal to both:
For vectors v = v, v, v and w = w, w, w:
v w = vw - vw, vw - vw, vw - vw
Geometric interpretation:
|v w| = |v| |w| sin()
Where is the angle between the vectors
Important properties:
- v w is orthogonal to both v and w
- v w = -w v (Not commutative)
- v w = 0 if and only if v is parallel to w
- i j = k, j k = i, k i = j
Position Vectors
A position vector represents the position of a point relative to the origin:
For a point P(x, y), the position vector OP:
OP = x, y = xi + yj
Work as a Dot Product
In physics, work can be calculated using the dot product:
W = F d = |F| |d| cos()
Where F is the force vector, d is the displacement vector, and is the angle between them
Distance Between Points Using Vectors
The distance between two points can be calculated using vectors:
For points P(x, y, z) and Q(x, y, z):
d(P, Q) = |PQ| = [(x - x) + (y - y) + (z - z)]
Important Vector Identities
For any vectors u, v, w and scalars a, b:
u + v = v + u (Commutative)
(u + v) + w = u + (v + w) (Associative)
(a + b)v = av + bv
a(u + v) = au + av
u (v + w) = u v + u w
(au) v = a(u v)
Applications of Vectors
Vectors have numerous applications in physics and engineering:
Forces and Motion: Describing forces, velocities, and accelerations
Navigation: Determining displacement and direction
Computer Graphics: Creating 3D models and animations
Electromagnetism: Describing electric and magnetic fields
Structural Engineering: Analyzing forces in structures
Fluid Dynamics: Modeling flow and pressure
Example: A force F = 3, 4 N moves an object from point (1, 2) to point (4, 6). Calculate the work done.
Solution: The displacement vector d = 4-1, 6-2 = 3, 4
Work = F d = 3(3) + 4(4) = 9 + 16 = 25 Joules
Solving Vector Problems
Here's a step-by-step approach to solving vector problems:
Identify the given information and what you need to find
Draw a diagram if applicable
Express vectors in component form if they aren't already
Apply the appropriate vector operations
Check your answer by verifying vector properties
Example: Find a vector of length 7 in the same direction as v = 3, 4
Solution:
1. Find the unit vector in the direction of v
2. |v| = (3 + 4) = 5
3. Unit vector = v/|v| = 3/5, 4/5
4. Multiply by 7: 73/5, 4/5 = 21/5, 28/5
Common Mistakes to Avoid
Confusing direction angle with bearing
Forgetting to adjust the angle for the correct quadrant
Applying dot or cross product formulas incorrectly
Dividing by zero when finding unit vectors or projections
Mixing up vector and scalar quantities
Forgetting the negative sign when reversing the order of vectors in a cross product
Mastery of these vector concepts will give you a solid foundation for more advanced topics in calculus, physics, and engineering.
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