This HL mini-lesson covers Vectors: components & magnitude, addition and scalar multiplication, the scalar (dot) product and angle between vectors, the vector (cross) product and area, and the vector equation of a line.
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Components
Vectors, components & magnitude
A vector has magnitude and direction, written in components. Its magnitude (length) in three dimensions is:
?Two non-zero vectors are perpendicular exactly when:
Algebra
Addition, scaling & unit vectors
Add vectors componentwise; a scalar k multiplies each component. A unit vector has magnitude 1:
â = a / |a|direction of a, length 1
Calculate
Magnitude
✎Find the magnitude of the vector (2, 3, 6).
Hint: √(2² + 3² + 6²) = √49.
Dot product
The scalar (dot) product
The dot product returns a number:
a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ(1,2,3)·(4,−5,6) = 4 − 10 + 18 = 12
Key test: a · b = 0 exactly when the vectors are perpendicular.
Quick check
Magnitude
?The magnitude of a = (a₁, a₂, a₃) is:
Calculate
Dot product
✎Find (1, 2, 3) · (4, −5, 6).
Hint: 1×4 + 2×(−5) + 3×6.
Angle
Angle between vectors
Rearranging the dot product gives the angle:
cos θ = (a · b) / (|a||b|)e.g. (1,0,0) and (1,1,0): cos θ = 1/√2, so θ = 45°
Sort it
Dot product, cross product or neither?
Tap a task, then the tool for it.
• Scalar (dot) product
× Vector (cross) product
➕ Neither
Cross product
The vector (cross) product & area
The cross product returns a vector perpendicular to both a and b, with magnitude:
|a × b| = |a||b| sin θ= area of the parallelogram spanned by a and b
The triangle formed by a and b has area ½|a × b|.
Calculate
Angle
✎Find the angle in degrees between (1, 0, 0) and (1, 1, 0).
°
Hint: cos θ = (a·b)/(|a||b|) = 1/√2.
Quick check
Cross product
?The vector a × b is:
Line equation
Vector equation of a line
A line through point A with direction b is the set of position vectors:
r = a + λba is a point on the line, b is its direction, λ ∈ ℝ
Calculate
Parallelogram area
✎Find the area of the parallelogram spanned by (2, 0, 0) and (0, 3, 0), i.e. |a × b|.
Hint: a × b = (0, 0, 6), so |a × b| = 6.
Match it
Match each statement to its meaning
Tap a statement on the left, then its matching partner on the right.
Statement
Meaning
Intersections
Angles & intersections of lines
The angle between two lines is the angle between their direction vectors (use the dot product). Lines intersect if a common point solves both equations; otherwise they are parallel or skew.
Calculate
Perpendicular test
✎Find (2, −1, 3) · (1, 5, 1) (a zero result means perpendicular).
Hint: 2×1 + (−1)×5 + 3×1.
Quick check
Line equation
?A vector equation of a line is r = a + λb, where b is:
Strategy
Choosing the product
Use the dot product for angles and perpendicularity; use the cross product for a perpendicular direction or an area. For lines, read off a point and a direction to build r = a + λb.
Exam habit: to test perpendicularity, just check the dot product is zero.
Recap
The big ideas to know
Magnitude: |a| = √(a₁²+a₂²+a₃²); unit â = a/|a|
Dot product: a·b = Σaᵢbᵢ = |a||b|cosθ; zero ⇒ perpendicular
Angle: cos θ = (a·b)/(|a||b|)
Cross product: |a×b| = |a||b|sinθ = parallelogram area
Line: r = a + λb (point a, direction b)
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