What are the products (either vector or scalar for each answer) of:
a) scalar x scalar
b) scalar x vector
c) vector x vector
A simple concept, but I want to be sure I get it right! Thanks for any help :)
Sure! Let's break down each scenario:
a) Scalar x Scalar:
When you multiply two scalar (single value) quantities, the product is also a scalar. For example, if you multiply the scalar values 3 and 4, the product would be 12, which is a scalar.
b) Scalar x Vector:
When you multiply a scalar by a vector, the result is a scaled vector in the direction of the original vector. For example, if you multiply the scalar value 2 by the vector (1, 2, 3), the result would be the scaled vector (2, 4, 6).
c) Vector x Vector:
When you multiply two vectors, the result is a vector that is perpendicular (orthogonal) to both input vectors, known as the cross product. The magnitude of the cross product vector represents the area of the parallelogram formed by the two input vectors, and the direction of the cross product vector is determined by the right-hand rule. Its direction is perpendicular to the plane formed by the two input vectors.
In this case, the dimensions of the input vectors should be compatible for the cross product. Specifically, the vectors must be three-dimensional for a cross product to be defined.
So, in summary:
a) scalar x scalar → scalar product
b) scalar x vector → scaled vector
c) vector x vector → cross product (vector)
I hope that clears up your understanding! If you have any further questions, feel free to ask.
You're welcome! I'll be happy to help you understand the products of scalar and vector quantities.
a) Scalar x Scalar: The product of two scalars (numbers that have magnitude only) is also a scalar. It is obtained by simply multiplying the two scalars together. For example, if you multiply the scalars 2 and 3, the product would be 2 x 3 = 6.
b) Scalar x Vector: When a scalar is multiplied by a vector, the result is a new vector. This new vector has the same direction as the original vector but has a magnitude that is scaled by the scalar. The magnitude of the original vector is multiplied by the scalar. For example, if you have the scalar 3 and the vector (1, 2, 3), the product would be 3 x (1, 2, 3) = (3, 6, 9).
c) Vector x Vector: The product of two vectors is called the vector cross product. The result of the cross product is a new vector that is perpendicular to both of the original vectors. The magnitude of the result vector is equal to the product of the magnitudes of the two original vectors multiplied by the sine of the angle between them. The direction of the resultant vector is given by the right-hand rule. The cross product is denoted by the symbol "x". For example, if you have the vectors A = (2, 1, 3) and B = (4, -2, 1), the cross product A x B would yield a new vector C = (-5, 10, -10).