State whether the following expressions using the cross product and dot product are vectors, scalars, or meaningless.
a•(b x c)
9 answers
Scalar.
Why?
The dot product of a vector and the cross product of two vectors will always result in a scalar quantity. This is because the cross product of two vectors results in a vector that is perpendicular to both vectors, and the dot product of a vector with a perpendicular vector is always zero. Therefore, the dot product in this expression only considers the magnitude of the vector, which is a scalar quantity.
how about (a+b)xc
Vector.
Why?
This is because the cross product of two vectors results in a vector, and when you add two vectors, the result is also a vector. Therefore, (a+b) is a vector, and when it is crossed with c, the result is also a vector. The cross product of two vectors always results in a vector that is perpendicular to both of the original vectors, and this vector has both magnitude and direction, so it is a true vector quantity.
What about (a x b)•(b x c)
Why?
Why?
Scalar.
The dot product of two vectors is a scalar. In this case, a x b and b x c are both vectors, but when you take their dot product, it results in a scalar quantity. This dot product only considers the multiplication of the magnitudes of two vectors and the angle between them, which can be expressed as a scalar. Therefore, (a x b) • (b x c) is a scalar quantity.
The dot product of two vectors is a scalar. In this case, a x b and b x c are both vectors, but when you take their dot product, it results in a scalar quantity. This dot product only considers the multiplication of the magnitudes of two vectors and the angle between them, which can be expressed as a scalar. Therefore, (a x b) • (b x c) is a scalar quantity.
0 answers