Key takeaway: The dot product multiplies two vectors into a single number by pairing components, and it equals the product of their lengths times the cosine of the angle between them.
Grade 12 – PrecalculusN-VM.B.5Geometry and Measurement
Definition of Dot Product
The dot product combines two vectors and returns a plain number, not another vector. Pair the first components, pair the second components, multiply each pair and add the results.
The number carries geometric meaning. It is positive when the two arrows point generally the same way, negative when they point generally opposite ways, and zero exactly when they are perpendicular. Equivalently it measures how much one arrow reaches along the other, scaled by the lengths. This is why work in physics, force times displacement along the force, is a dot product.
Formula and notation
For \(\mathbf{u}=\langle u_1,u_2\rangle\) and \(\mathbf{v}=\langle v_1,v_2\rangle\):
\[\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2=\|\mathbf{u}\|\,\|\mathbf{v}\|\cos\theta.\]
Here \(\theta\) is the angle between the arrows, \(0^{\circ}\le\theta\le180^{\circ}\).
Key facts
- A result of zero means the two arrows are perpendicular, if neither is the zero arrow.
- The order does not matter: u · v and v · u give the same number.
- Dotting an arrow with itself gives the square of its length.
- The output is a scalar, which is why it is also called the scalar product.
Dot Product example
Find the dot product of \(\mathbf{u}=\langle 6,-1\rangle\) and \(\mathbf{v}=\langle 2,5\rangle\), and the angle between them.
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Multiply matching components and add: \(6\cdot 2+(-1)\cdot 5=12-5=7\).
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Lengths: \(\|\mathbf{u}\|=\sqrt{37}\approx 6.083\) and \(\|\mathbf{v}\|=\sqrt{29}\approx 5.385\).
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Solve for the cosine: \(\cos\theta=\frac{7}{\sqrt{37}\sqrt{29}}\approx 0.2137\), so \(\theta\approx 77.7^{\circ}\).
The dot product is 7, and the arrows meet at about 77.7°.
Common mistake
Watch out
Mina multiplies the components and keeps them separate, writing \(\mathbf{u}\cdot\mathbf{v}=\langle 12,-5\rangle\).
Why it is wrong
The final step is to add the products, which collapses the pair into one number: \(12+(-5)=7\). A dot product is a scalar, never an arrow.
Related terms and lessons
Browse the glossary: A–Z list · Precalculus vocabulary · Geometry and Measurement