If vectors v ⃗ = ⟨ 4 , 3 ⟩ v ⃗=⟨4,3⟩ v ⃗ = ⟨ 4 , 3 ⟩ and u ⃗ = ⟨ 9 , 1 ⟩ u ⃗=⟨9,1⟩ u ⃗ = ⟨ 9 , 1 ⟩ , calculate v ⃗ ⋅ u ⃗ v ⃗⋅u ⃗ v ⃗ ⋅ u ⃗ .

⟨ 36 , 3 ⟩ \langle36,3\rangle ⟨ 36 , 3 ⟩

⟨ 4 , 27 ⟩ \langle4,27\rangle ⟨ 4 , 27 ⟩

If vectors ∣ v ⃗ ∣ = 12 |v ⃗ |=12 ∣ v ⃗∣ = 12 , ∣ u ⃗ ∣ = 100 |u ⃗ |=100 ∣ u ⃗∣ = 100 and the angle between v ⃗ v ⃗ v ⃗ & u ⃗ u ⃗ u ⃗ is θ = π 6 \theta=\frac{\pi}{6} θ = 6 π , calculate v ⃗ ⋅ u ⃗ v ⃗⋅u ⃗ v ⃗ ⋅ u ⃗ .

600 3 600\sqrt3 600 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

300 3 300\sqrt3 300 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

8. Vectors

Dot Product

8. Vectors

Dot Product: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

The dot product of two vectors is calculated by multiplying their corresponding components and summing the results, yielding a scalar that indicates their alignment. The formula can also be expressed using magnitudes and the cosine of the angle between them: $v^{2} = | v | | u | cos (θ)$ . A positive result indicates alignment, a negative result indicates opposition, and zero indicates perpendicularity.

concept

Introduction to Dot Product

Video duration:

Introduction to Dot Product Video Summary

In vector mathematics, one important operation is the dot product, which allows us to multiply two vectors together. The dot product is calculated by multiplying the corresponding components of the vectors and then summing those products. For example, if we have two vectors, v = (1, 2) and u = (3, 2), the dot product is computed as follows:

Dot Product: v · u = (1 × 3) + (2 × 2) = 3 + 4 = 7.

The result of the dot product is a scalar, which provides insight into the directional relationship between the two vectors. A positive result indicates that the vectors are aligned in the same direction, while a negative result suggests they are oriented in opposite directions. For instance, if we calculate the dot product of v = (-2, 3) and w = (2, 1), we find:

Dot Product: v · w = (-2 × 2) + (3 × 1) = -4 + 3 = -1.

This negative result indicates that the vectors are pulling against each other. Conversely, if the dot product equals zero, it signifies that the vectors are perpendicular, meaning they do not influence each other directionally. For example, if we have u = (3, 0) and w = (0, 4), the calculation would be:

Dot Product: u · w = (3 × 0) + (0 × 4) = 0 + 0 = 0.

This zero result confirms that the vectors are orthogonal, or perpendicular, to each other. Understanding these relationships through the dot product is crucial in various applications, including physics and engineering, where vector alignment can affect outcomes significantly.

Problem

If vectors $v ⃗=⟨4,3⟩$ and $u ⃗=⟨9,1⟩$ , calculate $v ⃗⋅u ⃗$ .

$39$

$\langle36,3\rangle$

$33$

$\langle4,27\rangle$

Problem

If vectors $v⃗=12î$ and $u⃗=100ĵ$ , calculate $u ⃗⋅v ⃗$ .

$1200k̂$

$1200$

$0$

$100$

Problem

If vectors $a⃗=13î$ , $⃗b⃗=5î-12ĵ$ , and $c⃗=24ĵ$ , calculate $b ⃗⋅(a ⃗-c ⃗)$ .

$353$

$132$

$65$

$-223$

concept

Find the Angle Between Vectors

Video duration:

Find the Angle Between Vectors Video Summary

In vector mathematics, the dot product is a crucial operation that allows us to multiply two vectors. Traditionally, this is done by multiplying their corresponding components and summing the results. However, there is an alternative method that utilizes the magnitudes of the vectors and the angle between them, which can be particularly useful in various applications.

The dot product can be expressed using the formula:

\[\mathbf{v} \cdot \mathbf{u} = |\mathbf{v}| \cdot |\mathbf{u}| \cdot \cos(\theta)\]

In this equation, $|\mathbf{v}|$ and $|\mathbf{u}|$ represent the magnitudes of vectors $\mathbf{v}$ and $\mathbf{u}$, respectively, and $\theta$ is the angle between the two vectors. This formulation is particularly advantageous when the component forms of the vectors are not readily available, or when the angle is known.

For example, if the magnitude of vector $\mathbf{v}$ is $\sqrt{5}$ and the magnitude of vector $\mathbf{u}$ is $\sqrt{13}$, with an angle of $29.7^\circ$ between them, the dot product can be calculated as follows:

\[\mathbf{v} \cdot \mathbf{u} = \sqrt{5} \cdot \sqrt{13} \cdot \cos(29.7^\circ)\]

By simplifying, we find:

\[\mathbf{v} \cdot \mathbf{u} = \sqrt{65} \cdot \cos(29.7^\circ)\]

Calculating this yields a result of approximately $7$, confirming that both methods of calculating the dot product yield the same result.

Moreover, this alternative dot product formula is instrumental in finding the smallest angle between two vectors. For instance, if the dot product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is given as $16$, and their magnitudes are $4$ and $8$ respectively, we can set up the equation:

\[16 = 4 \cdot 8 \cdot \cos(\theta)\]

From this, we simplify to find:

\[32 \cdot \cos(\theta) = 16\]

Dividing both sides by $32$ gives:

\[\cos(\theta) = \frac{16}{32} = \frac{1}{2}\]

Taking the inverse cosine of $\frac{1}{2}$ results in:

\[\theta = 60^\circ\]

This demonstrates how the dot product formula can be rearranged to solve for the angle between two vectors, showcasing its versatility in vector analysis.

Problem

If vectors $|a⃗|=3$ and $|b⃗|=7$ , and $a⃗\cdot b⃗=14.85$ , determine the angle between vectors $a ⃗$ and $b ⃗$ .

$21\degree$

$1.0\degree$

$45\degree$

$14.85\degree$

Problem

If vectors $a⃗=4î$ and $b⃗=3î-2ĵ$ , determine the angle between vectors $a ⃗$ and $b ⃗$ .

$46.10\degree$

$13.90\degree$

$43.90\degree$

$33.69\degree$

Problem

If vectors $|v ⃗ |=12$ , $|u ⃗ |=100$ and the angle between $v ⃗$ & $u ⃗$ is $\theta=\frac{\pi}{6}$ , calculate $v ⃗⋅u ⃗$ .

$600$

$600\sqrt3$

$300\sqrt3$

$300$

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We have more practice problems on Dot Product

Here’s what students ask on this topic:

What is the dot product of two vectors?

The dot product of two vectors is a scalar value obtained by multiplying their corresponding components and summing the results. For vectors $v$ and $u$ with components $v$ ₁, v₂) and $u$ ₁, u₂), the dot product is calculated as:

$v · u = v$ ₁u₁ + v₂u₂

This operation results in a scalar that indicates the degree of alignment between the two vectors.

How do you calculate the dot product using magnitudes and the angle between vectors?

The dot product can also be calculated using the magnitudes of the vectors and the cosine of the angle between them. For vectors $v$ and $u$ with magnitudes $|v|$ and $|u|$ , and angle $θ$ between them, the dot product is given by:

$v · u = |v||u|cos(θ)$

This formula is useful for finding the smallest angle between two vectors or when the vectors are given in terms of their magnitudes and the angle between them.

What does a positive, negative, or zero dot product indicate about the vectors?

A positive dot product indicates that the vectors are aligned in the same direction. A negative dot product indicates that the vectors are aligned in opposite directions. A zero dot product indicates that the vectors are perpendicular to each other, meaning they have no alignment.

How can you find the angle between two vectors using the dot product?

To find the angle $θ$ between two vectors $v$ and $u$ , you can rearrange the dot product formula:

$cos(θ) = \frac{v · u}{|v||u|}$

Then, take the inverse cosine (arccos) of both sides to solve for $θ$ :

$θ = arccos(\frac{v · u}{|v||u|})$

This will give you the smallest angle between the two vectors.

What is the geometric interpretation of the dot product?

Geometrically, the dot product of two vectors represents the product of their magnitudes and the cosine of the angle between them. It measures how much one vector extends in the direction of another. If the dot product is positive, the vectors point in a similar direction; if negative, they point in opposite directions; if zero, they are perpendicular.

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Patrick Ford

Physics and Math Lead Instructor

Dot Product: Videos & Practice Problems

Introduction to Dot Product

Introduction to Dot Product Video Summary

If vectors v⃗=⟨4,3⟩v ⃗=⟨4,3⟩v⃗=⟨4,3⟩ and u⃗=⟨9,1⟩u ⃗=⟨9,1⟩u⃗=⟨9,1⟩, calculate v⃗⋅u⃗v ⃗⋅u ⃗v⃗⋅u⃗.

If vectors v⃗=12ı^v⃗=12îv⃗=12ı^ and u⃗=100ȷ^u⃗=100ĵu⃗=100ȷ^, calculate u⃗⋅v⃗u ⃗⋅v ⃗u⃗⋅v⃗.

If vectors a⃗=13ı^a⃗=13îa⃗=13ı^, ⃗b⃗=5ı^−12ȷ^⃗b⃗=5î-12ĵ⃗b⃗=5ı^−12ȷ^, and c⃗=24ȷ^c⃗=24ĵc⃗=24ȷ^, calculate b⃗⋅(a⃗−c⃗)b ⃗⋅(a ⃗-c ⃗)b⃗⋅(a⃗−c⃗).

Find the Angle Between Vectors

Find the Angle Between Vectors Video Summary

If vectors ∣a⃗∣=3|a⃗|=3∣a⃗∣=3 and ∣b⃗∣=7|b⃗|=7∣b⃗∣=7, and a⃗⋅b⃗=14.85a⃗\cdot b⃗=14.85a⃗⋅b⃗=14.85, determine the angle between vectors a⃗a ⃗a⃗ and b⃗b ⃗b⃗.

If vectors a⃗=4ı^a⃗=4îa⃗=4ı^ and b⃗=3ı^−2ȷ^b⃗=3î-2ĵb⃗=3ı^−2ȷ^, determine the angle between vectors a⃗a ⃗a⃗ and b⃗b ⃗b⃗.

If vectors ∣v⃗∣=12|v ⃗ |=12∣v⃗∣=12, ∣u⃗∣=100|u ⃗ |=100 ∣u⃗∣=100 and the angle between v⃗v ⃗v⃗ & u⃗u ⃗u⃗ is θ=π6\theta=\frac{\pi}{6}θ=6π, calculate v⃗⋅u⃗v ⃗⋅u ⃗v⃗⋅u⃗ .

Do you want more practice?

Here’s what students ask on this topic:

What is the dot product of two vectors?

How do you calculate the dot product using magnitudes and the angle between vectors?

What does a positive, negative, or zero dot product indicate about the vectors?

How can you find the angle between two vectors using the dot product?

What is the geometric interpretation of the dot product?

Your Trigonometry tutors

If vectors $v ⃗=⟨4,3⟩$ and $u ⃗=⟨9,1⟩$ , calculate $v ⃗⋅u ⃗$ .

If vectors $v⃗=12î$ and $u⃗=100ĵ$ , calculate $u ⃗⋅v ⃗$ .

If vectors $a⃗=13î$ , $⃗b⃗=5î-12ĵ$ , and $c⃗=24ĵ$ , calculate $b ⃗⋅(a ⃗-c ⃗)$ .

If vectors $|a⃗|=3$ and $|b⃗|=7$ , and $a⃗\cdot b⃗=14.85$ , determine the angle between vectors $a ⃗$ and $b ⃗$ .

If vectors $a⃗=4î$ and $b⃗=3î-2ĵ$ , determine the angle between vectors $a ⃗$ and $b ⃗$ .

If vectors $|v ⃗ |=12$ , $|u ⃗ |=100$ and the angle between $v ⃗$ & $u ⃗$ is $\theta=\frac{\pi}{6}$ , calculate $v ⃗⋅u ⃗$ .