Scalar Product and Projections
The scalar product a.b = |a||b| cos t gives a number, and the projection of a on b is a.b/|b|.
What is Scalar Product and Projections?
The scalar product a.b = |a||b| cos t gives a number, and the projection of a on b is a.b/|b|.
Key formula / rule: a.b = 0 for non-zero vectors means they are perpendicular.
Key points
- Compute the angle between two vectors using the scalar product.
- Apply the scalar product to find projections and test perpendicularity.
Common exam trap
Reporting the dot product as a vector.
Definitions
- Term
Scalar Product and Projections
- Meaning
The scalar product a.b = |a||b| cos t gives a number, and the projection of a on b is a.b/|b|.
- Term
Scalar Product and Projections — explanation
- Meaning
The dot product measures alignment, so it is the natural test for perpendicularity and the tool for extracting components along a direction. Work done by a force is exactly this product.
Learning objectives
Compute the angle between two vectors using the scalar product.
Apply the scalar product to find projections and test perpendicularity.
Formulae
- Key point
a.b = 0 for non-zero vectors means they are perpendicular.
- Key point
a.a = |a|^2.
- Key point
The dot product is commutative and distributes over addition.
Prerequisites
MAT-U4-VA-T1-S1-C1
Common mistakes
Reporting the dot product as a vector.
Dividing by |a| instead of |b| when projecting a on b.
Keywords
Scalar
Product
Projections
Practice preview
If vector a = 2i + 3j - k and vector b = i - 2j + 4k, what is their scalar product a.b?…
easy
Find the projection of vector a = i + 2j + 3k on vector b = 2i + j - 2k.…
easy
What is the scalar product (dot product) of two non-zero vectors a and b, with an angle theta between them?…
easy