8.3 Vectors in a Cartesian Plane
In a Cartesian coordinate plane, any vector can be described using standard unit vectors along the coordinate axes or expressed as a column vector. This topic covers representing position vectors and displacement vectors in the forms \(x\underset{\sim}{i} + y\underset{\sim}{j}\) and \(\begin{pmatrix} x \\ y \end{pmatrix}\), calculating their magnitudes using the Pythagorean theorem, determining unit vectors in given directions, conducting algebraic operations, and applying these principles to solve real-world problems involving kinematics and navigation.
Learning Objectives
- Represent vectors on a Cartesian plane using unit vectors \(\underset{\sim}{i}\) and \(\underset{\sim}{j}\) or column vector form \(\begin{pmatrix} x \\ y \end{pmatrix}\).
- Determine the magnitude of vectors in the Cartesian plane using the formula \(|\underset{\sim}{r}| = \sqrt{x^2 + y^2}\).
- Describe and determine the unit vector in the direction of a given vector using \(\hat{\underset{\sim}{r}} = \dfrac{\underset{\sim}{r}}{|\underset{\sim}{r}|} = \dfrac{x\underset{\sim}{i} + y\underset{\sim}{j}}{\sqrt{x^2 + y^2}}\).
- Perform arithmetic operations involving vectors: addition, subtraction, scalar multiplication and combined operations.
- Solve real-world and geometric problems involving vectors in the Cartesian plane.
Representation and Components of a Vector in Cartesian Plane
Move the point \(B(x, y)\) to see how the position vector \(\vec{OB} = x\underset{\sim}{i} + y\underset{\sim}{j}\) and its magnitude change.
Position Vector \(\vec{OB}\) and Its Components
Explore the point \(B(x, y)\)
The values match the textbook example: \(x = 4\) and \(y = 3\).
Both components are positive.
Short Explanation
Vectors in a Cartesian coordinate system provide an algebraic way to describe directed line segments.
1. Unit Vectors Parallel to Axes
Vector of magnitude \(1\text{ unit}\) parallel to the positive \(x\text{-axis}\) is \(\underset{\sim}{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\), with \(|\underset{\sim}{i}| = 1\). Vector of magnitude \(1\text{ unit}\) parallel to the positive \(y\text{-axis}\) is \(\underset{\sim}{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\), with \(|\underset{\sim}{j}| = 1\).
2. Position Vector Relative to Origin \(O\)
If point \(B\) has coordinates \((x, y)\), the position vector of \(B\) relative to origin \(O\) is \(\vec{OB} = x\underset{\sim}{i} + y\underset{\sim}{j} = \begin{pmatrix} x \\ y \end{pmatrix}\).
3. Magnitude of a Vector
The magnitude of \(\vec{OB}\) or \(\underset{\sim}{r} = x\underset{\sim}{i} + y\underset{\sim}{j}\) is given by \(|\vec{OB}| = |\underset{\sim}{r}| = \sqrt{x^2 + y^2}\).
4. Unit Vector in the Direction of a Vector
If \(\underset{\sim}{r} = x\underset{\sim}{i} + y\underset{\sim}{j}\), the unit vector in the direction of \(\underset{\sim}{r}\) is \(\hat{\underset{\sim}{r}} = \dfrac{\underset{\sim}{r}}{|\underset{\sim}{r}|} = \dfrac{x\underset{\sim}{i} + y\underset{\sim}{j}}{\sqrt{x^2 + y^2}}\).
5. Arithmetic Operations on Vectors
Addition and subtraction are performed by grouping the \(\underset{\sim}{i}\) and \(\underset{\sim}{j}\) components separately. Scalar multiplication multiplies both components by the scalar.
6. Parallel Vectors
Two vectors \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\) are parallel if and only if one is a scalar multiple of the other: \(\underset{\sim}{a} = k\underset{\sim}{b}\), or if their gradients are equal: \(\dfrac{y_1}{x_1} = \dfrac{y_2}{x_2}\).
| Operation | Column vector form | Unit vector form |
|---|---|---|
| Addition | \(\begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}\) | \((a_1\underset{\sim}{i} + a_2\underset{\sim}{j}) + (b_1\underset{\sim}{i} + b_2\underset{\sim}{j}) = (a_1 + b_1)\underset{\sim}{i} + (a_2 + b_2)\underset{\sim}{j}\) |
| Subtraction | \(\begin{pmatrix} a_1 \\ a_2 \end{pmatrix} - \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 - b_1 \\ a_2 - b_2 \end{pmatrix}\) | \((a_1\underset{\sim}{i} + a_2\underset{\sim}{j}) - (b_1\underset{\sim}{i} + b_2\underset{\sim}{j}) = (a_1 - b_1)\underset{\sim}{i} + (a_2 - b_2)\underset{\sim}{j}\) |
| Scalar multiplication | \(k\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}\) | \(k(x\underset{\sim}{i} + y\underset{\sim}{j}) = kx\underset{\sim}{i} + ky\underset{\sim}{j}\) |
Magnitude of a vector
\(|\underset{\sim}{r}| = \sqrt{x^2 + y^2}\)Apply the Pythagorean theorem to the components.
Unit vector in the direction of \(\underset{\sim}{r}\)
\(\hat{\underset{\sim}{r}} = \dfrac{\underset{\sim}{r}}{|\underset{\sim}{r}|}\)Divide the vector by its magnitude to obtain a vector of length 1.
Try to Answer First
Answer in your mind, then press "Check Answer".
Given the coordinates of point \(P\) are \((-4, 3)\), write the position vector \(\vec{OP}\) in the form \(x\underset{\sim}{i} + y\underset{\sim}{j}\) and find its magnitude \(|\vec{OP}|\).
Find the unit vector in the direction of vector \(\underset{\sim}{a} = 3\underset{\sim}{i} + 2\underset{\sim}{j}\).
Given \(\underset{\sim}{p} = \begin{pmatrix} 7 \\ -1 \end{pmatrix}\) and \(\underset{\sim}{q} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}\), determine the vector \(\underset{\sim}{p} - \underset{\sim}{q}\).
Common Mistakes
- Squaring negative components incorrectly: writing \((-4)^2 = -16\) instead of \(+16\) when finding magnitude, leading to negative values under the square root.
- Mixing up unit vector components: adding the \(x\text{-component}\) of one vector to the \(y\text{-component}\) of another vector instead of grouping \(\underset{\sim}{i}\) with \(\underset{\sim}{i}\) and \(\underset{\sim}{j}\) with \(\underset{\sim}{j}\).
- Omitting the vector tilde or hat notation: writing \(i\) and \(j\) instead of \(\underset{\sim}{i}\) and \(\underset{\sim}{j}\), or writing \(r\) instead of unit vector \(\hat{\underset{\sim}{r}}\) or vector \(\underset{\sim}{r}\).
- Confusing column vector orientation: writing \(\begin{pmatrix} y \\ x \end{pmatrix}\) instead of \(\begin{pmatrix} x \\ y \end{pmatrix}\), placing the vertical component on top.
Concept Misunderstandings
A unit vector is only \(\underset{\sim}{i}\) or \(\underset{\sim}{j}\).
Any vector that has a magnitude equal to \(1\text{ unit}\), in any direction, is a unit vector.
Vector addition is the addition of magnitudes: \(|\underset{\sim}{a} + \underset{\sim}{b}| = |\underset{\sim}{a}| + |\underset{\sim}{b}|\).
Magnitudes cannot simply be added arithmetically unless the vectors are parallel and in the same direction.
Negative vector components imply negative magnitude.
Magnitude represents length/distance and is strictly non-negative (\(|\underset{\sim}{r}| \ge 0\)), regardless of whether \(x\) or \(y\) are negative numbers.
Summary
- Basic Unit Vectors: \(\underset{\sim}{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) (horizontal, positive \(x\)), \(\underset{\sim}{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\) (vertical, positive \(y\)).
- Position Vector: For \(A(x, y)\), \(\vec{OA} = x\underset{\sim}{i} + y\underset{\sim}{j} = \begin{pmatrix} x \\ y \end{pmatrix}\).
- Magnitude: \(|\underset{\sim}{r}| = \sqrt{x^2 + y^2}\).
- Unit Vector Formula: \(\hat{\underset{\sim}{r}} = \dfrac{\underset{\sim}{r}}{|\underset{\sim}{r}|} = \dfrac{x\underset{\sim}{i} + y\underset{\sim}{j}}{\sqrt{x^2 + y^2}}\).
- Arithmetic Rules: Addition/Subtraction add or subtract corresponding components separately; Scalar Multiplication multiplies each component by the scalar factor.
- Parallel Condition: \(\underset{\sim}{a} = k\underset{\sim}{b}\) or gradients \(\dfrac{y_a}{x_a} = \dfrac{y_b}{x_b}\).
Short Activity
Objective: Test your understanding of vector representation, magnitude, unit vectors and arithmetic operations.
A. Objective Quiz
1 What is the magnitude of the vector \(\underset{\sim}{u} = -12\underset{\sim}{i} - 5\underset{\sim}{j}\)?
2 Given that \(\underset{\sim}{a} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}\) and \(\underset{\sim}{b} = \begin{pmatrix} 4 \\ -12 \end{pmatrix}\), find \(2\underset{\sim}{a} - \underset{\sim}{b}\).
B. Fill in the Blanks
3 A vector with a magnitude of 1 unit parallel to the positive x-axis is written as column vector ______.
4 If the vector \(\begin{pmatrix} k \\ \frac{1}{\sqrt{2}} \end{pmatrix}\) is a unit vector, the positive value of \(k\) is ______.
C. Matching / Drag and Drop
Match each vector or operation on the left with its correct equivalent form or value on the right. On a phone, tap a term first, then tap the matching box.
