7 min read
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.
Choose Your Level
Addition, Subtraction, Scalar Multiples
What is it?
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.
In simple words
- 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
Important terms
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|.
What you should learn
- 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.
Examples
- (3i+j)+(i-4j)=4i-3j
- u-v = u+(-v)
- Triangle law picture
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|.
Recommended for Class 10
Full school-level explanation
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.
Like with like:
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.
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.
Remember
- component-wise
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|.
Remember
- |ku|=|k||u|
Examples
- 2u doubles length
- -u opposite
- k=0 gives 0
Remember
- add components
- negative k flips
- 2j
- no vector + scalar
- component-wise
- |ku|=|k||u|
- add components
- negative k flips
How the ideas connect
Worked: u = i + j, v = i - j. Find u - v. u - v = (1-1)i + (1-(-1))j = 0 i + 2 j = 2j
Deeper understanding
Go beyond memorising Addition, Subtraction, Scalar Multiples. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: u = i + j, v = i - j. Find u - v. u - v = (1-1)i + (1-(-1))j = 0 i + 2 j = 2j
Common mistakes
Exam focus
Core Idea
SUM AND DIFFERENCE
SCALAR k
Worked Method
Exam focus
Quick revision
- add components
- negative k flips
- 2j
- no vector + scalar
- 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 stre
- SUM AND DIFFERENCE: u+v is the diagonal of the parallelogram. u-v = u+(-v). Grade 10 examples stay in components: (1i+1j
You cannot add a vector to a scalar. 3 + (2i) is not defined in this chapter.
For long answers on Addition, Subtraction, Scalar Multiples: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- add components
- negative k flips
- 2j
- no vector + scalar
Important formulas
- add components
- negative k flips
- 2j
Common exam traps
- You cannot add a vector to a scalar. 3 + (2i) is not defined in this chapter.
- Do not treat SUM AND DIFFERENCE and SCALAR k as the same step; each has a different purpose.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
You cannot add a vector to a scalar. 3 + (2i) is not defined in this chapter.
- Mistake
Do not treat SUM AND DIFFERENCE and SCALAR k as the same step; each has a different purpose.
- Mistake
Do not jump straight to the answer; omitted signs, brackets, restrictions, or construction reasons can change the result.
- Mistake
You cannot add a vector to a scalar. 3 + (2i) is not defined in this chapter.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
add components
negative k flips
2j
no vector + scalar
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 parallelo
|ku|=|k||u|
- 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 parallelo
- Formula
- |ku|=|k||u|
- Example
- (3i+j)+(i-4j)=4i-3j
- Critical warning
- You cannot add a vector to a scalar. 3 + (2i) is not defined in this chapter.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.
Ready to test yourself?
You've completed the learning section.
Exam Practice
Define
Q1. Define Addition, Subtraction, Scalar Multiples.
Explain
Q2. Explain: SUM AND DIFFERENCE.
Example
Q3. Give one clear example of Addition, Subtraction, Scalar Multiples and say why it fits.
Mistake
Q4. State one common mistake students make about Addition, Subtraction, Scalar Multiples.
Exam note
Q5. Write a short exam note on Addition, Subtraction, Scalar Multiples: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.