1. REVIEW: INTRODUCTION TO VECTORS
Definition:
1.A vector is a quantity having both magnitude and direction.
2.Examples: displacement, velocity, acceleration, force.
Scalar vs Vector:
| Quantity | Magnitude | Direction | Examples |
|---|---|---|---|
| Scalar | Yes | No | Mass, Time, Distance, Temperature |
| Vector | Yes | Yes | Displacement, Velocity, Force, Acceleration |
Direction in Real Life:
1.Airplane flying: velocity has direction and speed.
2.Landing approach: aligns with runway direction.
3.Vector applications: wind flow, river current, GPS navigation.
Position Vector :
1.A position vector represents the location of a point relative to the origin.
2.For point A(x, y), position vector from origin O → OA = xi + yj
Unit Vectors :
1.Unit vectors are vectors of magnitude 1 pointing along coordinate axes.
2.i → along x-axis, |i| = 1
3.j → along y-axis, |j| = 1
4.Used to express vectors in component form
Any Vector :
1.Any vector in 2D can be expressed as a combination of unit vectors:
a = xi + yj
2.x and y are components along x- and y-axes respectively
3.Magnitude → |a| = √(x² + y²)
4.Direction → angle with x-axis θ, tan θ = y/x
2. TYPES OF VECTORS
1.Zero Vector → Magnitude = 0, coordinates (0, 0), no fixed direction.
2.Unit Vector → Magnitude = 1, examples: i, j.
3.Equal Vectors → Same magnitude and same direction.
4.Collinear Vectors → Parallel or anti-parallel vectors.
5.Negative Vector → Same magnitude, opposite direction.
6.Position Vector → Vector from origin to a point.
Applications of Vectors:
1.Navigation and GPS
2.Physics (forces, motion)
3.Engineering structures
4.Games and computer graphics
3. SCALAR (DOT) PRODUCT
Definition:
1.For vectors a and b with angle θ → a · b = |a||b| cos θ
2.Result is scalar.
Geometrical Meaning:
1.a · b = magnitude of a × projection of b on a.
2.If θ = 90° → cos 90° = 0 → a · b = 0 → vectors perpendicular.
Dot Product of Unit Vectors:
1.i · i = 1, j · j = 1
2.i · j = 0, j · i = 0
Component Form:
1.a = (x₁, y₁), b = (x₂, y₂) → a · b = x₁x₂ + y₁y₂
Magnitude of Vector:
1.|a|² = a · a = x₁² + y₁² → |a| = √(x₁² + y₁²)
Angle Between Two Vectors:
1.cos θ = (a · b)/(|a||b|)
2.Maximum when θ = 0°, minimum when θ = 180°
Properties of Dot Product:
1.a · b = b · a
2.a · (b + c) = a · b + a · c
3.(a + b)² = a² + 2 a · b + b²
4.(a − b)² = a² − 2 a · b + b²
5.(a + b)(a − b) = a² − b²
4. VECTOR GEOMETRY
Midpoint Formula:
1.Midpoint of A(a) and B(b) → (a + b)/2
Section Formula:
| Type | Position Vector of P |
|---|---|
| Internal Division (m:n) | (m b + n a)/(m + n) |
| External Division (m:n) | (m b − n a)/(m − n) |
Centroid of Triangle:
1.Vertices with vectors a, b, c → G = (a + b + c)/3
5. THEOREMS ON TRIANGLES
1.Midpoint Theorem → Line joining midpoints of two sides is parallel to third side and half its length.
2.Isosceles Triangle → Median to base is perpendicular to base.
3.Right Triangle → Midpoint of hypotenuse equidistant from all vertices.
6. QUADRILATERALS AND SEMICIRCLE
1.Varignon’s Theorem → Joining midpoints of quadrilateral forms a parallelogram.
2.Parallelogram → Diagonals bisect each other.
3.Rhombus → Diagonals bisect at 90°.
4.Rectangle → Diagonals are equal.
5.Semicircle Theorem → Angle in semicircle = 90°.
7. IMPORTANT FORMULAS
| Concept | Formula |
|---|---|
| Dot Product | a · b = x₁x₂ + y₁y₂ = |
| Magnitude | |
| Angle | cos θ = (a · b)/( |
| Midpoint | M = (a + b)/2 |
| Section (Internal) | P = (m b + n a)/(m + n) |
| Section (External) | P = (m b − n a)/(m − n) |
| Centroid | G = (a + b + c)/3 |
8. IMPORTANT QUESTIONS WITH SOLUTIONS
Question 1: Define scalar product
Solution:
or a · b = |a||b| cos θ → result is scalar
Question 2: If a · b = 0, what can you conclude?
Solution:
or Vectors a and b are perpendicular
Question 3: Find dot product of a = (2,3) and b = (4,−1)
Solution:
or a · b = 2×4 + 3×(−1)
or a · b = 8 − 3 = 5
Question 4: Find magnitude of vector a = (5,12)
Solution:
or |a| = √(5² + 12²)
or |a| = √(25 + 144) = √169 = 13
Question 5: Find angle between a = (1,2) and b = (2,1)
Solution:
or a · b = 1×2 + 2×1 = 4
or |a| = √(1² + 2²) = √5, |b| = √(2² + 1²) = √5
or cos θ = 4/(√5 × √5) = 4/5
or θ = cos⁻¹(4/5)
Question 6: Find midpoint of A(1,3) and B(5,7)
Solution:
or Midpoint = ((1 + 5)/2, (3 + 7)/2) = (3,5)
Question 7: Find position vector dividing A(2,4) and B(8,10) internally in 1:2
Solution:
or P = (1×B + 2×A)/(1+2) = ((8,10) + (4,8))/3 = (12,18)/3 = (4,6)
Question 8: Check if a = (3,4), b = (6,8) are parallel
Solution:
or b = 2a → Vectors are parallel
Question 9: Prove diagonals of parallelogram bisect each other
Solution:
or Midpoints of both diagonals using position vectors → midpoints are same → proved
Question 10: Right triangle → midpoint of hypotenuse equidistant from all vertices
Solution:
or Midpoint M of hypotenuse
or Distances MA, MB, MC = √[(x−xₘ)² + (y−yₘ)²] → equal for all vertices → proved
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https://besidedegree.com/exam/s/academic