Dot Product Math Example 4

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

Example 4

hard
Find โŸจ2,โˆ’1,3โŸฉโ‹…โŸจ4,5,โˆ’2โŸฉ\langle 2, -1, 3 \rangle \cdot \langle 4, 5, -2 \rangle.

Solution

  1. 1
    2(4)+(โˆ’1)(5)+3(โˆ’2)=8โˆ’5โˆ’6=โˆ’32(4) + (-1)(5) + 3(-2) = 8 - 5 - 6 = -3.
  2. 2
    Negative dot product means the angle between them is greater than 90ยฐ.

Answer

โˆ’3-3
The dot product formula extends to any dimension: multiply corresponding components and sum. The sign tells you about the angle โ€” negative means obtuse (> 90ยฐ).

About Dot Product

The dot product of two vectors a=โŸจa1,a2โŸฉ\mathbf{a} = \langle a_1, a_2 \rangle and b=โŸจb1,b2โŸฉ\mathbf{b} = \langle b_1, b_2 \rangle is the scalar aโ‹…b=a1b1+a2b2\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2. Equivalently, aโ‹…b=โˆฅaโˆฅโˆฅbโˆฅcosโกฮธ\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta, where ฮธ\theta is the angle between the vectors.

Learn more about Dot Product โ†’

More Dot Product Examples

Example 1 easy

Find [formula].

Example 2 medium

Find the angle between [formula] and [formula].

Example 3 easy

Are [formula] and [formula] perpendicular?

โ† All Dot Product Examples Dot Product Concept