9. Vectors

Dot Product

9. Vectors

Dot Product: Videos & Practice Problems

Video Lessons Worksheet

Concept

Introduction to Dot Product

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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.

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Problem

If vectors $v ⃗ equals left angle bracket 4 comma 3 right angle bracket$ and $u ⃗ equals left angle bracket 9 comma 1 right angle bracket$ , calculate $v ⃗ \cdot u ⃗$ .

$39$

$left angle bracket 36 comma 3 right angle bracket$

$33$

$left angle bracket 4 comma 27 right angle bracket$

Problem

If vectors $v ⃗ equals 12 dotless i hat$ and $u ⃗ equals 100 dotless j hat$ , calculate $u ⃗ \cdot v ⃗$ .

$1200 k hat$

$1200$

$0$

$100$

Problem

If vectors $a ⃗ equals 13 dotless i hat$ , $⃗ b ⃗ = 5 \hat{ı} - 12 \hat{ȷ}$ , and $c ⃗ equals 24 dotless j hat$ , calculate $b ⃗ \cdot (a ⃗ - c ⃗)$ .

$353$

$132$

$65$

$negative 223$

Concept

Find the Angle Between Vectors

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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 $∣ a ⃗ ∣ = 3$ and $∣ b ⃗ ∣ = 7$ , and $a ⃗ \cdot b ⃗ = 14.85$ , determine the angle between vectors $a ⃗$ and $b ⃗$ .

$21 degrees$

$1.0 degrees$

$45 degrees$

$14.85 degrees$

Problem

If vectors $a ⃗ equals 4 dotless i hat$ and $b ⃗ = 3 \hat{ı} - 2 \hat{ȷ}$ , determine the angle between vectors $a ⃗$ and $b ⃗$ .

$46.10 degrees$

$13.90 degrees$

$43.90 degrees$

$33.69 degrees$

Problem

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

$600$

$600$\sqrt$3$

$300$\sqrt$3$

$300$

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Dot Product: Videos & Practice Problems

Introduction to Dot Product

Introduction to Dot Product Video Summary

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

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

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

Find the Angle Between Vectors

Find the Angle Between Vectors Video Summary

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

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

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

Your Algebra and Trigonometry tutors

If vectors $v ⃗ equals left angle bracket 4 comma 3 right angle bracket$ and $u ⃗ equals left angle bracket 9 comma 1 right angle bracket$ , calculate $v ⃗ \cdot u ⃗$ .

If vectors $v ⃗ equals 12 dotless i hat$ and $u ⃗ equals 100 dotless j hat$ , calculate $u ⃗ \cdot v ⃗$ .

If vectors $a ⃗ equals 13 dotless i hat$ , $⃗ b ⃗ = 5 \hat{ı} - 12 \hat{ȷ}$ , and $c ⃗ equals 24 dotless j hat$ , calculate $b ⃗ \cdot (a ⃗ - c ⃗)$ .

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

If vectors $a ⃗ equals 4 dotless i hat$ and $b ⃗ = 3 \hat{ı} - 2 \hat{ȷ}$ , 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 $v ⃗⋅u ⃗$ .