Question

For the two vectors \(\vec{A}=(2,1,0)\) and \(\vec{B}=(0,1,2),\) what is their vector product, \(\vec{A} \times \vec{B} ?\) a) (2,-4,2) c) (2,0,2) e) (0,0,0) b) (1,0,1) d) (3,-2,1)

Short answer

Answer: The vector product of \(\vec{A}\) and \(\vec{B}\) is (2,-4,2).

Step-by-step solution

Recall the formula for the cross product of two vectors.

The vector product, or cross product, of two vectors \(\vec{A}\) and \(\vec{B}\) is given by the following formula: \(\vec{A} \times \vec{B} = \left(\begin{bmatrix}A_y B_z - A_z B_y \\ A_z B_x - A_x B_z\\ A_x B_y - A_y B_x\end{bmatrix}\right).\) Now we can plug in the given values for \(\vec{A}=(2,1,0)\) and \(\vec{B}=(0,1,2)\) to find the cross product.

Plug in the given values and compute the cross product.

Using the formula from the previous step, we can now compute the cross product \(\vec{A} \times \vec{B}\): \(\vec{A} \times \vec{B} = \left(\begin{bmatrix}1(2) - 0(1) \\ 0(0) - 2(2) \\ 2(1) - 1(0) \end{bmatrix}\right) = \left(\begin{bmatrix}2 \\ -4 \\ 2 \end{bmatrix}\right).\) After computing the cross product, we obtain the vector \((2,-4,2)\).

Match the computed vector with the given options.

Now, we can match the computed vector \((2,-4,2)\) with the given options to find the correct answer. The option that matches is: a) (2,-4,2) Therefore, the vector product \(\vec{A} \times \vec{B}\) is \((2,-4,2)\).

Key concepts

Cross Product

The cross product, also known as the vector product, is a fundamental operation in vector calculus used extensively in physics and engineering. Unlike the dot product, which provides a scalar quantity, the cross product of two vectors results in a new vector. It's particularly useful in determining a vector that is perpendicular to the plane formed by the initial two vectors.

For any two vectors \(\vec{A}\) and \(\vec{B}\), the cross product \(\vec{A} \times \vec{B}\) is calculated using the formula:
\[\vec{A} \times \vec{B} = \left(\begin{bmatrix}A_y B_z - A_z B_y \ A_z B_x - A_x B_z\ A_x B_y - A_y B_x\end{bmatrix}\right).\]
Using this operation, we can also determine the area of a parallelogram formed by two vectors or the torque resulting from a force applied at a point in space. Importantly, the direction of the resulting vector is given by the right-hand rule, which aids in visualizing and ensuring the proper orientation in three-dimensional space.

Vectors in Physics

Vectors play an integral role in physics as they represent quantities that have both magnitude and direction. Common examples include displacement, velocity, acceleration, and force. Understanding vectors is crucial in solving problems related to motion and statics where directionality influences the system's behavior.

When analyzing physical phenomena, it's essential to correctly add, subtract, and multiply vectors. These operations follow specific rules that take into account both the magnitude and direction of the vectors involved. For example, the cross product can represent a rotational effect in physical scenarios, such as calculating the moment of a force about a pivot point, which is a vector quantity. Knowledge of how to manipulate vectors mathematically enables physicists and engineers to model and predict behaviors in the physical world with accuracy.

Vector Calculation

Vector calculation involves operations including addition, subtraction, dot product, and cross product. It's the groundwork for more complex vector algebra and calculus used in various scientific disciplines. For instance, in graphics programming, cross products are used to compute normals to surfaces, which are then used for lighting calculations.

Let's improve the understanding of cross product calculation with an example. Given vectors \(\vec{A}=(2,1,0)\) and \(\vec{B}=(0,1,2)\), the cross product \(\vec{A} \times \vec{B}\) is found by:
\[\vec{A} \times \vec{B} = \left(\begin{bmatrix}1(2) - 0(1) \ 0(0) - 2(2) \ 2(1) - 1(0) \end{bmatrix}\right) = \left(\begin{bmatrix}2 \ -4 \ 2 \end{bmatrix}\right).\]
This resultant vector can then be compared to given options, as seen in the exercise provided. Remember to maintain the order of the components when performing the calculation to ensure the accuracy of the result. In teaching vector calculations, practical examples and visual aids can greatly enhance comprehension for students.