Calculating Dot Product Using Components: Videos & Practice Problems

The dot product of two vectors can be calculated using their components by multiplying the parallel components together. For vectors expressed in unit vector form, the dot product is given by the equation: $a^b=(a{x}_{}*b{x}_{})+(a{y}_{}*b{y}_{})+(a{z}_{}*b{z}_{})$. This method is essential when magnitudes and angles are not provided, yielding a scalar result that reflects the relationship between the vectors.

Calculating Dot Product Using Vector Components

The dot product of two vectors can be calculated using either their magnitudes and the angle between them or their components. When using magnitudes, the formula is given by:

$$ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) $$

Here, \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are the magnitudes of vectors \( \mathbf{a} \) and \( \mathbf{b} \), and \( \theta \) is the angle between them. This approach focuses on the multiplication of the parallel components of the vectors.

However, when vectors are expressed in terms of their components, such as \( \mathbf{a} = 2\mathbf{i} + 3\mathbf{j} \) and \( \mathbf{b} = \mathbf{i} + 2\mathbf{j} \), the dot product can be calculated using the formula:

$$ \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z $$

In this case, \( a_x, a_y, a_z \) are the components of vector \( \mathbf{a} \) in the \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) directions, respectively, and \( b_x, b_y, b_z \) are the components of vector \( \mathbf{b} \). The key is to multiply the corresponding components and then sum the results.

For example, if we calculate the dot product of \( \mathbf{a} = 2\mathbf{i} + 3\mathbf{j} \) and \( \mathbf{b} = \mathbf{i} + 2\mathbf{j} \), we find:

$$ \mathbf{a} \cdot \mathbf{b} = (2)(1) + (3)(2) = 2 + 6 = 8 $$

This result is a scalar, which is characteristic of the dot product.

In another example, consider \( \mathbf{a} = -3\mathbf{i} + 0\mathbf{j} + 4\mathbf{k} \) and \( \mathbf{b} = 1\mathbf{i} - 2\mathbf{j} \). The calculation would proceed as follows:

$$ \mathbf{a} \cdot \mathbf{b} = (-3)(1) + (0)(-2) + (4)(0) = -3 + 0 + 0 = -3 $$

In this case, the absence of a \( k \) component in vector \( \mathbf{b} \) means that the \( k \) term contributes zero to the dot product. Thus, the final result is simply the sum of the products of the corresponding components.

Understanding how to compute the dot product using both methods is essential for solving various problems in physics and engineering, as it provides insight into the relationship between vectors in different contexts.

Calculate the dot product between A = (6.6 i - 3.4 j - 6.4 k ) and B = (8.6 i + 2.6 j - 5.8 k).

• A. 85
• B. 15 i - 0.80 j - 12 k
• C. 11
• D. 57 i - 8.8 j + 37 k

Answer: (A)

Calculating the Angle Between 2 Vectors Using the Dot Product

In this example, we explore the calculation of the dot product and the angle between two vectors, a and b, represented in unit vector components. The dot product, denoted as a · b, can be calculated using the formula that involves the components of the vectors. For vectors expressed in terms of their unit components, the dot product is computed by pairing the corresponding components and summing the products. Specifically, if a is given as 7.2i - 3.9j and b as 2.1i + 4.8j, the dot product is calculated as follows:

a · b = (7.2)(2.1) + (-3.9)(4.8)

Calculating this gives:

a · b = 15.12 - 18.72 = -3.6

This result indicates that the dot product is -3.6, which can be interpreted as a measure of how aligned the two vectors are; a negative value suggests they point in somewhat opposite directions.

Next, we determine the angle θ between the two vectors. The relationship between the dot product and the angle is given by the equation:

a · b = |a| |b| cos(θ)

To find θ, we first need to calculate the magnitudes of both vectors. The magnitude of a vector can be found using the Pythagorean theorem:

|a| = √(7.2² + (-3.9)²) = √(51.84 + 15.21) = √67.05 ≈ 8.2

|b| = √(2.1² + 4.8²) = √(4.41 + 23.04) = √27.45 ≈ 5.2

Now, substituting the values into the dot product equation:

-3.6 = (8.2)(5.2) cos(θ)

Solving for cos(θ) gives:

cos(θ) = -3.6 / (8.2 * 5.2) ≈ -0.08

To find the angle θ, we take the inverse cosine:

θ = cos^-1(-0.08) ≈ 94.6°

This angle indicates that the vectors are nearly perpendicular, as expected from the negative cosine value. A sketch of the vectors confirms their orientation, with vector a having a positive x-component and a negative y-component, while vector b has both positive components, resulting in an angle slightly greater than 90 degrees.

Here’s what students ask on this topic:

How do you calculate the dot product of two vectors using their components?

To calculate the dot product of two vectors using their components, you multiply the corresponding components of the vectors and then sum the results. For vectors $a$ and $b$ with components $\mathrm{a\_x}$, $\mathrm{a\_y}$, $\mathrm{a\_z}$ and $\mathrm{b\_x}$, $\mathrm{b\_y}$, $\mathrm{b\_z}$, the dot product is given by:

$a\cdot b=\mathrm{a\_x}\cdot \mathrm{b\_x}+\mathrm{a\_y}\cdot \mathrm{b\_y}+\mathrm{a\_z}\cdot \mathrm{b\_z}$

This method is useful when magnitudes and angles are not provided, and it yields a scalar result.

What is the formula for the dot product of two vectors in component form?

The formula for the dot product of two vectors in component form is:

$a\cdot b=\mathrm{a\_x}\cdot \mathrm{b\_x}+\mathrm{a\_y}\cdot \mathrm{b\_y}+\mathrm{a\_z}\cdot \mathrm{b\_z}$

Here, $\mathrm{a\_x}$, $\mathrm{a\_y}$, $\mathrm{a\_z}$ are the components of vector $a$, and $\mathrm{b\_x}$, $\mathrm{b\_y}$, $\mathrm{b\_z}$ are the components of vector $b$. This formula allows you to find the dot product by multiplying the corresponding components and summing the results.

Why is the dot product of two vectors a scalar?

The dot product of two vectors is a scalar because it represents the sum of the products of their corresponding components. When you multiply the components of two vectors and add them together, the result is a single number, not a vector. This scalar value reflects the magnitude of the projection of one vector onto another, indicating how much one vector extends in the direction of the other.

Can you calculate the dot product if one vector is in 2D and the other in 3D?

Yes, you can calculate the dot product if one vector is in 2D and the other in 3D by treating the missing component of the 2D vector as zero. For example, if $a$ is a 2D vector with components $\mathrm{a\_x}$ and $\mathrm{a\_y}$, and $b$ is a 3D vector with components $\mathrm{b\_x}$, $\mathrm{b\_y}$, and $\mathrm{b\_z}$, you can consider $\mathrm{a\_z}$ as zero. The dot product is then:

$a\cdot b=\mathrm{a\_x}\cdot \mathrm{b\_x}+\mathrm{a\_y}\cdot \mathrm{b\_y}+\mathrm{a\_z}\cdot \mathrm{b\_z}$

Since $\mathrm{a\_z}$ is zero, the term $\mathrm{a\_z}\cdot \mathrm{b\_z}$ will be zero, simplifying the calculation.

What is the significance of the dot product in physics?

The dot product is significant in physics because it quantifies the extent to which two vectors align with each other. It is used in various applications, such as calculating work done by a force, where the dot product of the force vector and displacement vector gives the work. It also helps in determining the angle between vectors and in projecting one vector onto another, which is essential in analyzing vector components in different directions.