MathematicsChapter 3: Vectors
Angles and Applications (school)
Angle between two vectors: use the dot (cos) unless the question asks for sin (then |cross|).
Learning objectives
- State the main ideas and notation used in angles and applications (school).
- Apply the rules of angles and applications (school) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Angle between two vectors: use the dot (cos) unless the question asks for sin (then |cross|). Angle between a vector and a plane, or two planes, is a Grade 12-style extra — only if the book asks. Work W = F · d. Area of triangle ABC is (12)|AB x AC|. Meaning: pick dot for 'how aligned', pick cross for 'how much area / a perpendicular'.
Mathematical form
AB=<2,0,0>, AC=<0,2,0>
area = (12)*4 = 2
W = F
AC = 4 k area
Where / symbols
- ANGLES AND APPLICATIONS (SCHOOL)
- Angle between two vectors: use the dot (cos) unless the question asks for sin (then |cross|). Angle between a vector and a plane, or two planes, is a Grade 12-style extra — only if the book asks. Work W = F · d. Area of triangle ABC is (1/2)|AB x AC|. Meaning: pick dot for 'how aligned', pick cross for 'how much area / a perpendicular'.
- 1. WHICH PRODUCT
- Perp test: dot. Parallel test: cross=0 or components proportional. Unit normal to plane of a,b: a x b direction.
- 2. CLEAN ANGLES
- cos theta = (a·b)/(|a||b|). If the paper wants the acute angle, take the smaller of theta and 180-theta. Keep sqrt exact until the last line if asked.
Overview
Choose the product:
Which Product
Perp test: dot. Parallel test: cross=0 or components proportional. Unit normal to plane of a,b: a x b direction.
Examples F·d work · Area parallelogram |a x b| · Coplanar triple 0
Clean Angles
cos theta = a·b|a||b|. If the paper wants the acute angle, take the smaller of theta and 180-theta. Keep sqrt exact until the last line if asked.
Examples Not (a·b)/a·b · Both lengths · Calculator in degree mode if the answer is in degrees
WHAT — ANGLES AND APPLICATIONS (SCHOOL)
Angle between two vectors: use the dot (cos) unless the question asks for sin (then |cross|). Angle between a vector and a plane, or two planes, is a Grade 12-style extra — only if the book asks. Work W = F · d. Area of triangle ABC is (12)|AB x AC|. Meaning: pick dot for 'how aligned', pick cross for 'how much area / a perpendicular'.
RULE — 1. WHICH PRODUCT
Perp test: dot. Parallel test: cross=0 or components proportional. Unit normal to plane of a,b: a x b direction.
METHOD — 2. CLEAN ANGLES
cos theta = a·b|a||b|. If the paper wants the acute angle, take the smaller of theta and 180-theta. Keep sqrt exact until the last line if asked.
WORKED EXAMPLE
Worked triangle area A=<0,0,0>, B=<2,0,0>, C=<0,2,0>. AB=<2,0,0>, AC=<0,2,0> AB x AC = 4 k area = (12)*4 = 2
How to answer a angles and applications (school) question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- Worked triangle area A=<0,0,0>, B=<2,0,0>, C=<0,2,0>.
- AB=<2,0,0>, AC=<0,2,0>
- AB x AC = 4 k
- area = (12)*4 = 2
In simple words
Work is a scalar (dot). Torque as r x F is a vector (cross), named only.
Key points
- cos: dot
- area: |cross|
- dot vs cross
- divide by both moduli
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Angles and Applications (school)
Sticky Notes · Quick Revision
Core Idea
- Angle between two vectors: use the dot (cos) unless the question asks for sin (then |cross|).
- Angle between a vector and a plane, or two planes, is a Grade 12-style extra — only if the book asks.
- Work W = F · d.
1. WHICH PRODUCT
- Perp test: dot.
- Parallel test: cross=0 or components proportional.
- Unit normal to plane of a,b: a x b direction.
2. CLEAN ANGLES
- cos theta = a·b|a||b|.
- If the paper wants the acute angle, take the smaller of theta and 180-theta.
- Keep sqrt exact until the last line if asked.
Worked Example
- Worked triangle area A=<0,0,0>, B=<2,0,0>, C=<0,2,0>.
- AB=<2,0,0>, AC=<0,2,0>
- AB x AC = 4 k
- area = (12)*4 = 2
Common Misconception
- Work is a scalar (dot). Torque as r x F is a vector (cross), named only.
FBISE Exam Check
- State the rule for angles and applications (school) before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Angle between two vectors: use the dot (cos) unless the question asks for sin (then |cross|).
- Angle between a vector and a plane, or two planes, is a Grade 12-style extra — only if the book asks.
- Work W = F · d.
- Work is a scalar (dot). Torque as r x F is a vector (cross), named only.
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