A scalar quantity is a quantity that has only magnitude, with no direction associated with it. Here are some examples of scalar quantities that can be formed using vectors a and b:
Magnitude of a vector: The magnitude of a vector a is a scalar quantity, denoted by |a|. It is given by:
|a| = sqrt(a . a)
where a . a is the dot product of a with itself.
Dot product of two vectors: The dot product of two vectors a and b is a scalar quantity, denoted by a . b. It is given by:
a . b = |a| |b| cos(theta)
where |a| and |b| are the magnitudes of vectors a and b, respectively, and theta is the angle between them.
Projection of a vector: The projection of a vector a onto another vector b is a scalar quantity, denoted by proj_b(a). It is given by:
proj_b(a) = |a| cos(theta)
where theta is the angle between vectors a and b.
Scalar triple product of three vectors: The scalar triple product of three vectors a, b, and c is a scalar quantity, denoted by (a x b) . c. It is given by:
(a x b) . c = det([a b c])
where det([a b c]) is the determinant of the matrix formed by the vectors a, b, and c.