Find the angle between the vectors displaystyle<{1},{3}>{quadtext{and}quad}<-{2},{4}>

Question

Find the angle between the vectors

< 1, 3 > and < - 2, 4 >

Answer 1

Remember that the angle between two vectors u and v is defined by

\cos (θ) = \frac{u \cdot v}{| | u | | | | v | |}

So lets calculate

u \cdot v, | | u | |, and | | v | | f or u = ⟨ 1, 3 ⟩ and v = ⟨ - 2, 4 ⟩

u \cdot v = ⟨ 1, 3 ⟩ \cdot ⟨ - 2, 4 ⟩ = - 2 + 12 = 10

| | u | | = \sqrt{1^{2} + 3^{2}} = \sqrt{10}

| | v | | = \sqrt{{(- 2)}^{2} + 4^{2}} = \sqrt{20}

Now plugging these into our equation, we find that

\cos (θ) = \frac{10}{\sqrt{10} \sqrt{20}} = \frac{1}{\sqrt{2}}

Looking at the unit circle, we see that

\cos (θ) = \frac{1}{\sqrt{2}} a t 45^{\circ} and 315^{\circ}

. Because the angle between two vectors must be between

0^{\circ} and 180^{\circ}

, that means that the angle between vectors u and v is

45^{\circ}

Expert Answer