Multiple Choice
For two non-parallel vectors and , the cross product points in which direction?
62
views
3. Vectors
Intro to Cross Product (Vector Product)
3:22 minutesProblem 46a
Textbook Question
For the two vectors in Fig. E1.35, find the magnitude and direction of the vector product A x B
Verified step by step guidance1
First, understand that the vector product, also known as the cross product, of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by \( \mathbf{A} \times \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \sin(\theta) \mathbf{n} \), where \( \theta \) is the angle between the vectors and \( \mathbf{n} \) is a unit vector perpendicular to the plane containing \( \mathbf{A} \) and \( \mathbf{B} \).
Calculate the magnitudes of vectors \( \mathbf{A} \) and \( \mathbf{B} \) from the given lengths: \( |\mathbf{A}| = 2.80 \text{ cm} \) and \( |\mathbf{B}| = 1.90 \text{ cm} \).
Determine the angle \( \theta \) between the vectors \( \mathbf{A} \) and \( \mathbf{B} \). From the diagram, both vectors make an angle of \( 60.0^\circ \) with the x-axis, so the angle between them is \( 120.0^\circ \).
Substitute the values into the cross product formula: \( |\mathbf{A} \times \mathbf{B}| = (2.80 \text{ cm})(1.90 \text{ cm}) \sin(120.0^\circ) \). Use the fact that \( \sin(120.0^\circ) = \sin(60.0^\circ) = \frac{\sqrt{3}}{2} \).
The direction of the vector product \( \mathbf{A} \times \mathbf{B} \) is given by the right-hand rule. Point your fingers in the direction of \( \mathbf{A} \) and curl them towards \( \mathbf{B} \); your thumb will point in the direction of \( \mathbf{A} \times \mathbf{B} \), which is perpendicular to the plane containing \( \mathbf{A} \) and \( \mathbf{B} \). In this case, it will be out of the page.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Product (Cross Product)
The vector product, or cross product, of two vectors A and B, denoted as A x B, results in a third vector that is perpendicular to the plane formed by A and B. The magnitude of the cross product is given by |A||B|sin(θ), where θ is the angle between the two vectors. This operation is essential in physics for determining torque, angular momentum, and the magnetic force on a charged particle.
Recommended video:
Guided course
10:30
Vector (Cross) Product and the Right-Hand-Rule
Magnitude of a Vector
The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem in Cartesian coordinates. For a vector represented in two dimensions, the magnitude can be found using the formula |A| = √(Ax² + Ay²), where Ax and Ay are the vector's components along the x and y axes. Understanding the magnitude is crucial for calculating the vector product and interpreting physical quantities.
Recommended video:
Guided course
03:59Calculating Magnitude & Components of a Vector
Direction of a Vector
The direction of a vector indicates the orientation of the vector in space and is often expressed in terms of angles relative to a reference axis. In the context of the vector product, the direction of the resulting vector A x B is determined by the right-hand rule, which states that if the fingers of the right hand curl from vector A to vector B, the thumb points in the direction of A x B. This concept is vital for visualizing and solving problems involving vector interactions.
Recommended video:
Guided course
06:44Adding 3 Vectors in Unit Vector Notation
Related Videos
Related Practice