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Wize University Physics Textbook (Master) > Introduction to Vectors

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\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

If vector A is (1,2,3)\left(1,2,3\right) and vector B is (3,4,−5)\left(3,4,-5\right). Find the dot product of A and B.
Find the angle between the vectors i+j−ki+j-k and 2i−2j+k2i - 2j + k.