MathematicsChapter 7: Vectors in Plane
Addition, Subtraction, Scalar Multiples
Add component-wise: (x1 i + y1 j) + (x2 i + y2 j) = (x1+x2)i + (y1+y2)j.
Learning objectives
- Explain addition, subtraction, scalar multiples using the correct definitions, conditions, and notation.
- Apply add components, negative k flips correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
Add component-wise: (x1 i + y1 j) + (x2 i + y2 j) = (x1+x2)i + (y1+y2)j. Subtract likewise. k u stretches (or shrinks) by |k| and reverses if k<0. The parallelogram (or triangle) law is the picture of the same addition.
Mathematical form
u = i + j
v = i - j. Find u - v. u - v
j = 0 i + 2 j
Where / symbols
- ADDITION, SUBTRACTION, SCALAR MULTIPLES
- Add component-wise: (x1 i + y1 j) + (x2 i + y2 j) = (x1+x2)i + (y1+y2)j. Subtract likewise. k u stretches (or shrinks) by |k| and reverses if k<0. The parallelogram (or triangle) law is the picture of the same addition.
- SUM AND DIFFERENCE
- u+v is the diagonal of the parallelogram. u-v = u+(-v). Grade 10 examples stay in components: (1i+1j) - (1i-1j) = 2j.
- SCALAR k
- 2(3i-4j)=6i-8j. -OP for OP=[-6,7] is [6,-7]. Length scales by |k|: |k u| = |k| |u|.
Overview
Like with like:
Sum and Difference
u+v is the diagonal of the parallelogram. u-v = u+(-v). Grade 10 examples stay in components: (1i+1j) - (1i-1j) = 2j.
Examples (3i+j)+(i-4j)=4i-3j · u-v = u+(-v) · Triangle law picture
SCALAR k
2(3i-4j)=6i-8j. -OP for OP=[-6,7] is [6,-7]. Length scales by |k|: |k u| = |k| |u|.
Examples 2u doubles length · -u opposite · k=0 gives 0
WHAT — ADDITION, SUBTRACTION, SCALAR MULTIPLES
Add component-wise: (x1 i + y1 j) + (x2 i + y2 j) = (x1+x2)i + (y1+y2)j. Subtract likewise. k u stretches (or shrinks) by |k| and reverses if k<0. The parallelogram (or triangle) law is the picture of the same addition. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Addition, Subtraction, Scalar Multiples matters because vectors describe both size and direction, so they are useful for displacement, velocity, force, and geometry. Components turn a directional problem into manageable algebra.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through SUM AND DIFFERENCE, then SCALAR k. Worked: u = i + j, v = i - j. Find u - v. u - v = (1-1)i + (1-(-1))j = 0 i + 2 j = 2j Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- u = i + j, v = i - j. Find u - v.
- u - v = (1-1)i + (1-(-1))j
- = 0 i + 2 j
- = 2j
In simple words
You cannot add a vector to a scalar. 3 + (2i) is not defined in this chapter.
Key points
- add components
- negative k flips
- component-wise
- |ku|=|k||u|
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Addition, Subtraction, Scalar Multiples
Sticky Notes · Quick Revision
Core Idea
- Add component-wise: (x1 i + y1 j) + (x2 i + y2 j) = (x1+x2)i + (y1+y2)j. Subtract likewise. k u stretches (or shrinks) by |k| and reverses if k<0. The parallelogram (or triangle) law is the picture of the same addition.
SUM AND DIFFERENCE
- u+v is the diagonal of the parallelogram. u-v = u+(-v). Grade 10 examples stay in components: (1i+1j) - (1i-1j) = 2j.
SCALAR k
- 2(3i-4j)=6i-8j. -OP for OP=[-6,7] is [6,-7]. Length scales by |k|: |k u| = |k| |u|.
Worked Method
- Worked: u = i + j, v = i - j. Find u - v.
- u - v = (1-1)i + (1-(-1))j
- = 0 i + 2 j
- = 2j
Common Misconception
- You cannot add a vector to a scalar. 3 + (2i) is not defined in this chapter.
FBISE Exam Check
- You cannot add a vector to a scalar. 3 + (2i) is not defined in this chapter.
60-second revision
- Add component-wise: (x1 i + y1 j) + (x2 i + y2 j) = (x1+x2)i + (y1+y2)j.
- SUM AND DIFFERENCE: u+v is the diagonal of the parallelogram.
- SCALAR k: 2(3i-4j)=6i-8j.
- Worked-method reminder: = 2j
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.