Mathematics (AS)·Vectors in three dimensions · NSSCAS 1.7.2
Magnitude & the scalar product
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How long is a vector, and what is the angle between two of them? In this lesson you will measure any 3D vector with Pythagoras used twice, build unit vectors of length exactly one, and master the scalar (dot) product. Then you will use it to find angles to one decimal place and to prove that two vectors are perpendicular — including the classic exam questions with an unknown t and a right angle at a vertex.
What you'll learn in this lesson
By the end you should be able to (NSSCAS Mathematics (AS) 1.7.2):
- Calculate the magnitude of a vector in three dimensions using |a| = √(x² + y² + z²), and build unit vectors â = a/|a|
- Calculate the scalar product of two vectors: a·b = x₁x₂ + y₁y₂ + z₁z₂
- Use the scalar product to determine the angle between two directions with cos θ = a·b / (|a||b|)
- Use the scalar product to solve problems about perpendicularity of vectors (a·b = 0 ⇔ perpendicular)
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