Intro to Cross Product (Vector Product): Videos & Practice Problems

Vectors can be multiplied using two methods: the scalar (dot) product and the vector (cross) product. The scalar product yields a number, calculated as $a^b\cos \left(\theta \right)$, while the vector product results in a new vector, calculated as $a^b\sin (\mathrm{\theta <mo>)</mo>}$. The direction of the vector product is determined using the right-hand rule, indicating its perpendicular orientation to the original vectors.

Vector (Cross) Product and the Right-Hand-Rule

In vector mathematics, there are two primary methods for multiplying vectors: the scalar product (or dot product) and the vector product (or cross product). The scalar product results in a single number, while the vector product yields a new vector that is perpendicular to the original vectors involved in the multiplication.

The scalar product of two vectors, denoted as \( \mathbf{A} \cdot \mathbf{B} \), is calculated using the formula:

\( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta) \)

where \( |\mathbf{A}| \) and \( |\mathbf{B}| \) are the magnitudes of vectors \( \mathbf{A} \) and \( \mathbf{B} \), and \( \theta \) is the angle between them. For example, if \( |\mathbf{A}| = 3 \), \( |\mathbf{B}| = 4 \), and \( \theta = 60^\circ \), the scalar product would be:

\( 3 \times 4 \times \cos(60^\circ) = 6 \)

This value represents the multiplication of the parallel components of the vectors.

In contrast, the vector product, denoted as \( \mathbf{A} \times \mathbf{B} \), produces a new vector \( \mathbf{C} \) that is perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \). The magnitude of this vector is calculated using the formula:

\( |\mathbf{C}| = |\mathbf{A}| |\mathbf{B}| \sin(\theta) \)

Using the same vectors as before, if \( \theta = 60^\circ \), the magnitude of vector \( \mathbf{C} \) would be:

\( 3 \times 4 \times \sin(60^\circ) = 10.4 \)

To determine the direction of the vector product, the right-hand rule is employed. This involves pointing the fingers of your right hand along the first vector \( \mathbf{A} \) and curling them towards the second vector \( \mathbf{B} \). Your thumb will then point in the direction of the resulting vector \( \mathbf{C} \). If the thumb points towards you, the direction is represented by a dot (•), indicating it is coming out of the page. Conversely, if the thumb points away from you, it is represented by an 'X', indicating it is going into the page.

For example, if you have two vectors with magnitudes of 6 and 5 at an angle of 30 degrees, the magnitude of the vector product would be:

\( |\mathbf{C}| = 6 \times 5 \times \sin(30^\circ) = 15 \)

Using the right-hand rule, if you point your fingers along vector \( \mathbf{A} \) and curl them towards vector \( \mathbf{B} \), your thumb will point upwards, indicating that vector \( \mathbf{C} \) points along the positive Y-axis.

In another scenario, if two vectors are parallel, the angle between them is either 0 or 180 degrees, leading to a sine value of 0. Thus, the vector product in such cases results in a magnitude of 0, indicating no perpendicular vector exists.

Understanding these concepts is crucial for applications in physics and engineering, where vector operations are frequently utilized to analyze forces, motion, and other vector quantities.

Find the magnitude and direction of the vector C =2B × A.

• A. |C| = 27; along the –x direction
• B. |C| = 27; along the +x direction
• C. |C| = 54; along the +x direction
• D. |C| = 54; along the +y direction
• E. |C| = 19; along the +x direction

Answer: (C)

Find the Vector Product

To find the magnitude and direction of the cross product of two vectors, we can denote these vectors as **A** and **B**. The cross product, represented as **C = A × B**, results in a new vector **C** that is perpendicular to both **A** and **B**. The magnitude of the cross product can be calculated using the formula:

|C| = |A| |B| \sin(\theta)

In this equation, |A| and |B| are the magnitudes of vectors **A** and **B**, respectively, and θ is the angle between them. For example, if |A| = 12 and |B| = 8, we need to determine the sine of the angle between them. In a three-dimensional context, it can be tricky to visualize the angle. If **A** is horizontal and **B** is vertical, the angle between them is actually 90 degrees, even if it appears otherwise in a diagram.

Using the sine of 90 degrees, which is 1, we can calculate the magnitude of the cross product:

|C| = 12 × 8 × 1 = 96

Next, to determine the direction of vector **C**, we apply the right-hand rule. This involves pointing the fingers of your right hand in the direction of vector **A** and curling them towards vector **B**. The direction your thumb points indicates the direction of vector **C**. In this case, if **A** is tilted away from you and **B** is pointing straight up, your thumb will point off to the right, indicating the direction of **C**.

To find the angle θ that vector **C** makes with the x-axis, we can analyze the situation from a top-down perspective. If we know that the angle between the x-axis and vector **A** is 30 degrees, and since vector **C** is perpendicular to both **A** and **B**, we can deduce that the angle between the x-axis and vector **C** is also 30 degrees. Thus, the positive angle from the x-axis to vector **C** is:

θ = 30 degrees

In summary, the magnitude of the cross product **C** is 96, and it makes a positive angle of 30 degrees with the x-axis.

Finding Angle Using Scalar & Vector Products

To determine the angle θ between two vectors, A and B, we can utilize their scalar (dot) and vector (cross) products. The magnitude of the vector product is given by the equation:

$$ |C| = |A| |B| \sin(\theta) $$

In this case, we know that the magnitude of the vector product is 12, leading to:

$$ \sin(\theta) = \frac{12}{|A| |B|} $$

However, without the magnitudes of vectors A and B, we cannot solve for θ directly. We also have the scalar product defined as:

$$ A \cdot B = |A| |B| \cos(\theta) $$

Given that the scalar product is -8, we can express this as:

$$ \cos(\theta) = \frac{-8}{|A| |B|} $$

Now, we have two equations involving sine and cosine of θ, both dependent on the unknown magnitudes of A and B. To eliminate these magnitudes, we can divide the sine equation by the cosine equation:

$$ \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{12}{|A| |B|}}{\frac{-8}{|A| |B|}} $$

This simplifies to:

$$ \tan(\theta) = \frac{12}{-8} = -1.5 $$

To find θ, we take the inverse tangent:

$$ \theta = \tan^{-1}(-1.5) $$

Calculating this gives us approximately -56.3 degrees. However, since we are looking for the smallest positive angle between the vectors, we need to adjust this value. The negative angle indicates that the angle is measured in the clockwise direction, which is not suitable for our context.

Given that the scalar product is negative, we know that the angle between the vectors must be greater than 90 degrees. Therefore, we add 180 degrees to our result:

$$ \theta = -56.3 + 180 = 123.7 \text{ degrees} $$

This angle of 123.7 degrees is the correct answer, as it satisfies the conditions of both the scalar and vector products. Thus, the angle between vectors A and B is 123.7 degrees.

Here’s what students ask on this topic:

What is the difference between the dot product and the cross product of vectors?

The dot product (scalar product) of two vectors results in a scalar (a single number), calculated using the formula $ab\cos \theta$. The cross product (vector product), on the other hand, results in a new vector, calculated using the formula $ab\sin \theta$. The direction of the resulting vector from the cross product is perpendicular to the plane formed by the original vectors, determined using the right-hand rule.

How do you use the right-hand rule to determine the direction of the cross product?

To use the right-hand rule, follow these steps: 1) Point your fingers in the direction of the first vector (A). 2) Curl your fingers towards the direction of the second vector (B). 3) Your thumb will point in the direction of the resulting vector (C). If your thumb points out of the page, it is represented by a dot inside a circle. If it points into the page, it is represented by an 'X' inside a circle.

What is the formula for calculating the magnitude of the cross product of two vectors?

The magnitude of the cross product of two vectors A and B is given by the formula $\mathrm{|C|}=\mathrm{|A|}\mathrm{|B|}\sin \theta$, where $\mathrm{|A|}$ and $\mathrm{|B|}$ are the magnitudes of vectors A and B, and $\theta$ is the angle between them.

What happens to the cross product if the two vectors are parallel?

If two vectors are parallel, the angle between them is either 0 or 180 degrees. In both cases, the sine of the angle is zero ($\sin 0=0$ and $\sin 180=0$), making the magnitude of the cross product zero. Therefore, the cross product of two parallel vectors is a zero vector.

Can the cross product be used in three-dimensional space, and how is it represented?

Yes, the cross product can be used in three-dimensional space. It is represented by a new vector that is perpendicular to the plane formed by the original vectors. The direction of this vector is determined using the right-hand rule. In a 3D coordinate system, the cross product of vectors A and B can be calculated using the determinant of a matrix formed by the unit vectors i, j, k and the components of A and B.