Vectors and Components

Develop vector addition and scalar multiplication, magnitude and unit vectors.

Lesson 3 of 42 parts Practice Flashcards

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Vector Addition and Scalar Multiplication

Objective: Explain vector addition and scalar multiplication, apply the rule, and justify relevant restrictions.

A plane vector ⟨u₁,u₂⟩ represents displacement with both magnitude and direction. Add vectors componentwise and multiply both components by a scalar. A negative scalar reverses direction; a scalar magnitude scales length. Position does not alter a free vector: arrows with equal components represent the same displacement. A component is signed, whereas a magnitude is nonnegative.

Formulas

$u+v=⟨u₁+v₁,u₂+v₂⟩$

Worked examples

Try itFor u=⟨3,5⟩ and v=⟨7,−3⟩, find the x-component of u+v.Show answer

Step 1: Add corresponding components. Step 2: u+v=⟨10,2⟩. Step 3: The x-component is 10.

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Vector Addition and Scalar Multiplication

Objective: Explain vector addition and scalar multiplication, apply the rule, and justify relevant restrictions.

Formulas

$u+v=⟨u₁+v₁,u₂+v₂⟩$

Worked examples

Try itFor u=⟨3,5⟩ and v=⟨7,−3⟩, find the x-component of u+v.Show answer

Step 1: Add corresponding components. Step 2: u+v=⟨10,2⟩. Step 3: The x-component is 10.

Magnitude and Unit Vectors

Objective: Explain magnitude and unit vectors, apply the rule, and justify relevant restrictions.

The magnitude of v=⟨v₁,v₂⟩ is √(v₁²+v₂²), obtained from the Pythagorean theorem. For v≠0, its unit direction vector is v/|v|, which has length one. The zero vector has zero magnitude but no defined unit direction. A vector’s component sum does not generally equal its magnitude; signs indicate direction rather than negative length.

Formulas

$|v|=√(v₁²+v₂²); û=v/|v| for v≠0$

Worked examples

Try itFind the magnitude of v=⟨21,28⟩.Show answer

Step 1: Use |v|=√(vₓ²+vᵧ²). Step 2: The squared magnitude is 1225. Step 3: The magnitude is 35, a nonnegative scalar.

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