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Wize University Statics Textbook (Master) > Force Vectors

Dot Product

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DOT PRODUCT: aka "scalar product" as the answer is a scalar (#).

Can be used in two situations:
1) need to find the angle between 2 lines or vectors in 3D (in 2D use trig!!!)
2) find the projection of a vector onto a line (or pole or part of the structure)


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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\boxed{\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!


Alternatively, we can find the dot product between two vectors using their lengths as:

u⃗⋅v⃗=∣∣u⃗∣∣∣∣v⃗∣∣cos⁡θ\boxed{\vec{u}\cdot\vec{v}=\left|\left|\vec{u}\right|\right|\left|\left|\vec{v}\right|\right|\cos\theta}
where θ\theta is the angle between the two vectors


Wize Tip
Two non-zero vectors are perpendicular to each other if their dot product is equal to zero.



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Example: Dot Product of Two Vectors


a.) Given that u⃗=(1,4,−2)\vec{u}=\left(1,4,-2\right) and v⃗=(0,−2,5)\vec{v}=\left(0,-2,5\right), find u⃗\vec{u}.v⃗. \vec{v}.

(1)(0)+(4)(−2)+(−2)(5)(1)(0)+(4)(−2)+(−2)(5)
=−8−10=−8−10
=−18=−18

b.) If (2,−13, k) . (k, 9, −1)=5\left(2,-\frac{1}{3},\ k\right)\ .\ \left(k,\ 9,\ -1\right)=5, find the value of kk.

(2)(k)+(−13)(9)+(k)(−1)=5\left(2\right)\left(k\right)+\left(-\frac{1}{3}\right)\left(9\right)+\left(k\right)\left(-1\right)=5
2k−3−k=52k-3-k=5
k−3=5k-3=5
k=8k=8

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