Vector Addition, Subtraction, and Scalar Multiplication Math Example 1

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

Example 1

easy
If u=⟨3,−1⟩\mathbf{u} = \langle 3, -1 \rangle and v=⟨1,4⟩\mathbf{v} = \langle 1, 4 \rangle, find 2u−v2\mathbf{u} - \mathbf{v}.

Solution

  1. 1
    Step 1: 2u=⟨6,−2⟩2\mathbf{u} = \langle 6, -2 \rangle.
  2. 2
    Step 2: 2u−v=⟨6−1,−2−4⟩=⟨5,−6⟩2\mathbf{u} - \mathbf{v} = \langle 6 - 1, -2 - 4 \rangle = \langle 5, -6 \rangle.
  3. 3
    Check: Each component is 2ui−vi2u_i - v_i ✓

Answer

⟨5,−6⟩\langle 5, -6 \rangle
Scalar multiplication scales each component, then subtraction is done component-wise. Linear combinations of vectors like au+bva\mathbf{u} + b\mathbf{v} are fundamental in linear algebra.

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 2 medium

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

Example 3 medium

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

Example 4 easy

Compute [formula].

Example 5 medium

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

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