KKhaemenes PrecalculusTeacher Answer Key
Student Lesson

Teacher Key · Unit 09 · Lesson 04

Dot Product, Angle & Orthogonality

Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.

Khaemenes Academy · Teacher Key

Dot Product, Angle & Orthogonality

U09 · L04
1

Find the magnitude of ⟨3,4⟩.

Vectors visualRepresent magnitude and direction with vectors, resolve components, combine forces or motions, and use the dot product to analyze angle and orthogonality. xyvₓvᵧ
Answer: 5
Reasoning: Use √(3²+4²).
2

Write a vector of magnitude 10 at 30° in component form.

Vectors visualRepresent magnitude and direction with vectors, resolve components, combine forces or motions, and use the dot product to analyze angle and orthogonality. xyvₓvᵧ
Answer: ⟨5√3,5⟩
Reasoning: Use ⟨r cosθ,r sinθ⟩.
3

Add ⟨2,−1⟩ and ⟨5,4⟩.

Vectors visualRepresent magnitude and direction with vectors, resolve components, combine forces or motions, and use the dot product to analyze angle and orthogonality. xyvₓvᵧ
Answer: ⟨7,3⟩
Reasoning: Add corresponding components.
4

Compute ⟨1,2⟩·⟨3,−4⟩.

Vectors visualRepresent magnitude and direction with vectors, resolve components, combine forces or motions, and use the dot product to analyze angle and orthogonality. xyvₓvᵧ
Answer: −5
Reasoning: 1·3+2·(−4)=−5.
5

Are ⟨2,1⟩ and ⟨1,−2⟩ orthogonal?

Vectors visualRepresent magnitude and direction with vectors, resolve components, combine forces or motions, and use the dot product to analyze angle and orthogonality. xyvₓvᵧ
Answer: Yes
Reasoning: Their dot product is 0.
6

Find the unit vector in the direction of ⟨6,8⟩.

Vectors visualRepresent magnitude and direction with vectors, resolve components, combine forces or motions, and use the dot product to analyze angle and orthogonality. xyvₓvᵧ
Answer: ⟨3/5,4/5⟩
Reasoning: Divide by magnitude 10.
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