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Wize High School Grade 12 Calculus Textbook > Vector Products

Dot Product

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Dot Product

Given any two vectors u⃗=[u1, u2]\vec{u}=\left[u_1,\ u_2\right] and v⃗=[v1, v2]\vec{v}=\left[v_1,\ v_2\right] in R2R^2, the dot product is defined by
u⃗⋅v⃗=u1v1+u2v2\vec{u}\cdot\vec{v}=u_1v_1+u_2v_2
  • The dot product between two vectors is a scalar (number)!
  • The dot product between any two vectors in R3R^3 is calculated in a similar way [a1, a2, a3]⋅[b1, b2, b3]=a1b1+a2b2+a3b3\left[a_1,\ a_2,\ a_3\right]\cdot\left[b_1,\ b_2,\ b_3\right]=a_1b_1+a_2b_2+a_3b_3
Watch Out!
We can only calculate the dot product between two vectors that are in the same space!

Geometric Interpretation

u⃗⋅v⃗=∣∣u⃗∣∣∣∣v⃗∣∣cos⁡θ\vec{u}\cdot\vec{v}=\left|\left|\vec{u}\right|\right|\left|\left|\vec{v}\right|\right|\cos\theta where 0≤θ≤180°0\le\theta\le180\degree is the angle between the two vectors
❓ What does it mean if the dot product between two non-zero vectors is 0?
0=∣∣u⃗∣∣ ∣∣v⃗∣∣cos⁡θ0=\left|\left|\vec{u}\right|\right|\ \left|\left|\vec{v}\right|\right|\cos\theta
Since the two vectors are non-zero, we know that cos⁡θ=0\cos\theta=0
So, θ=90°\theta=90\degree → The two vectors are perpendicular (a.k.a. orthogonal or normal)



Write it Down
Any two non-zero vectors u⃗\vec{u} and v⃗\vec{v} are orthogonal (a.k.a. perpendicular or normal) if and only if u⃗⋅v⃗=0\vec{u}\cdot\vec{v}=0


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Example: Angle Between Vectors

Find the angle between the vectors u⃗=[1, −1]\vec{u}=\left[1,\ -1\right] and v⃗=[−5, 3]\vec{v}=\left[-5,\ 3\right].

u⃗⋅v⃗=∣∣u⃗∣∣∣∣v⃗∣∣cos⁡θ\vec{u}\cdot\vec{v}=\left|\left|\vec{u}\right|\right|\left|\left|\vec{v}\right|\right|\cos\theta
[1, −1]⋅[−5, 3]=∣∣[1, −1]∣∣∣∣[−5, 3]∣∣cos⁡θ\left[1,\ -1\right]\cdot\left[-5,\ 3\right]=\left|\left|\left[1,\ -1\right]\right|\right|\left|\left|\left[-5,\ 3\right]\right|\right|\cos\theta
−8=(1)2+(−1)2(−5)2+(3)2cos⁡θ-8=\sqrt{\left(1\right)^2+\left(-1\right)^2}\sqrt{\left(-5\right)^2+\left(3\right)^2}\cos\theta
−8=234cos⁡θ-8=\sqrt{2}\sqrt{34}\cos\theta
−8234=cos⁡θ-\frac{8}{\sqrt{2}\sqrt{34}}=\cos\theta
−868=cos⁡θ-\frac{8}{\sqrt{68}}=\cos\theta
−8417=cos⁡θ-\frac{8}{\sqrt{4}\sqrt{17}}=\cos\theta
−8217=cos⁡θ-\frac{8}{2\sqrt{17}}=\cos\theta
θ=cos⁡−1(−417)\theta=\cos^{-1}\left(-\frac{4}{\sqrt{17}}\right)
θ≈165.96\theta\approx165.96

Practice: Dot Product

Find the value(s) of kk such that the vectors v⃗=[1, k, −2]\vec{v}=\left[1,\ k,\ -2\right] and u⃗=[2k,k−1,k]\vec{u}=\left[2k,k-1,k\right] are perpendicular.
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Dot Product Properties

Suppose that u⃗\vec{u}, v⃗\vec{v}, w⃗\vec{w} are vectors in R2R^2 or R3R^3, and a, ba,\ b are scalars (numbers).
Justify the following properties.
  • Commutative Property: u⃗⋅v⃗=v⃗⋅u⃗\vec{u}\cdot\vec{v}=\vec{v}\cdot\vec{u}
  • Associative Property w/ a Scalar: a(u⃗⋅v⃗)=(au⃗)⋅v⃗=u⃗⋅(av⃗)a\left(\vec{u}\cdot\vec{v}\right)=\left(a\vec{u}\right)\cdot\vec{v}=\vec{u}\cdot\left(a\vec{v}\right)
  • Distributive Property:
  • u⃗⋅(v⃗+w⃗)=u⃗⋅v⃗+u⃗⋅w⃗\vec{u}\cdot\left(\vec{v}+\vec{w}\right)=\vec{u}\cdot\vec{v}+\vec{u}\cdot\vec{w}
  • (u⃗+v⃗)⋅w⃗=u⃗⋅w⃗+v⃗⋅w⃗\left(\vec{u}+\vec{v}\right)\cdot\vec{w}=\vec{u}\cdot\vec{w}+\vec{v}\cdot\vec{w}
  • Magnitude Property: u⃗⋅u⃗=∣∣u⃗∣∣2\vec{u}\cdot\vec{u}=\left|\left|\vec{u}\right|\right|^2

Practice: Dot Product Properties

Suppose that u⃗, v⃗, w⃗\vec{u},\ \vec{v},\ \vec{w} are vectors in R2R^2, p⃗, q⃗, r⃗\vec{p},\ \vec{q},\ \vec{r} are vectors in R3R^3 and a, ba,\ b are scalars.

Which of the following are valid operations? (Select all that apply)