Vector Addition, Subtraction, and Scalar Multiplication Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

medium
Given u = <2, -1> and v = <3, 5>, find 2u - v and its magnitude.

Solution

  1. 1
    Compute 2u2\mathbf{u}: 2⟨2,−1⟩=⟨4,−2⟩2\langle 2, -1 \rangle = \langle 4, -2 \rangle.
  2. 2
    Subtract v\mathbf{v}: ⟨4,−2⟩−⟨3,5⟩=⟨4−3,−2−5⟩=⟨1,−7⟩\langle 4, -2 \rangle - \langle 3, 5 \rangle = \langle 4-3, -2-5 \rangle = \langle 1, -7 \rangle.
  3. 3
    Find the magnitude: ∥⟨1,−7⟩∥=12+(−7)2=1+49=50=52≈7.07\|\langle 1, -7 \rangle\| = \sqrt{1^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2} \approx 7.07.

Answer

2u−v=⟨1,−7⟩2\mathbf{u} - \mathbf{v} = \langle 1, -7 \rangle, magnitude =52≈7.07= 5\sqrt{2} \approx 7.07
Linear combinations of vectors are computed component-wise: first scale, then add or subtract. The magnitude of the result uses the Pythagorean theorem applied to the components.

About Vector Addition, Subtraction, and Scalar Multiplication

Vectors are added and subtracted component by component. Scalar multiplication multiplies each component of a vector by a number. If u=⟨u1,u2⟩\mathbf{u} = \langle u_1, u_2 \rangle and v=⟨v1,v2⟩\mathbf{v} = \langle v_1, v_2 \rangle, then u+v=⟨u1+v1,u2+v2⟩\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle and ku=⟨ku1,ku2⟩k\mathbf{u} = \langle ku_1, ku_2 \rangle.

Learn more about Vector Addition, Subtraction, and Scalar Multiplication →

More Vector Addition, Subtraction, and Scalar Multiplication Examples

Example 1 easy

If [formula] and [formula], find [formula].

Example 2 medium

Find [formula] where [formula] and [formula].

Example 4 easy

Compute [formula].

Example 5 medium

Find the vector from point [formula] to point [formula].

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