Cogito
Precalculus · Chapter 3 · Lesson 2
The Dot Product
Multiply two vectors and get a number.
12 problems · about 22 minutes · TEKS P.4.H, TEKS P.4.I
What this lesson teaches
The student computes the dot product and uses it to find the angle between two vectors and to test for perpendicularity.
- The dot product multiplies matching components and adds, giving a number.
- It equals |a||b| cos θ, so it delivers the angle between two vectors.
- A dot product of zero means perpendicular.
Warm Up
Straightforward practice. Get the method working first.
5 problems(3, 2) · (1, 4)?
Answer 11
Why 11.
What does a dot product of zero mean?
Answer The vectors are perpendicular.
Why They meet at 90°.
(1, 2) · (3, 4)?
Answer 11
Why 3 + 8.
(5, 0) · (0, 7)?
Answer 0
Why Along the axes.
(2, 3) · (2, 3)?
Answer 13
Why 4 + 9, which is the magnitude squared.
Build It Up
The same ideas with more to keep track of.
3 problems(−2, 1) · (4, 2)?
Answer -6
Why −8 + 2.
Magnitude of (3, 4)?
Answer 5
Why √(9 + 16).
a · b = 0. What is the angle between them, in degrees?
Answer 90
Why cos θ = 0.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each pair by what the sign of its dot product tells you.
Answer Angle under 90°: Dot product = 12, Dot product = 0.5 · Angle over 90°: Dot product = −5, Dot product = −20
Why Cosine is positive below 90° and negative above it.
(4, 0) · (0, 9)?
Answer 0
Why Each product has a zero in it.
The Product: (2, 5) · (4, 1). What is the result?
Answer 13
Why 13.
The Right Angle: (6, 2) · (−1, 3). What is the result?
Answer 0
Why 0, so they are perpendicular.