The scalar product of vectors is used to find angles between vectors and in the definitions of derived scalar physical quantities such as work or energy. There are two kinds of multiplication for vectors. One kind of multiplication is the scalar product, also known as the dot product. The other kind of multiplication is the vector product, also known as the cross product. The scalar product of vectors is a number .

It even provides a simple test to determine whether two vectors meet at a right angle. Let be the vector with initial point and terminal point as shown in . Express in both component form and using standard unit vectors. Find the vector difference a−ba−b and express it in both the component form and by using the standard unit vectors.

As you might expect, to calculate the dot product of four-dimensional vectors, we simply add the products of the components as before, but the sum has four terms instead of three. Determine the vectors 2a,2a, −b,−b, and 2a−b.2a−b. Express the vectors in both the component form and by using standard unit vectors. The angles formed by a nonzero vector and the coordinate axes are called the direction angles for the vector ().

Two examples of cross products where the unit vectors do not appear in the cyclic order. Determine all three-dimensional vectors u orthogonal to vector Express the answer in component form. Find all two-dimensional vectors a orthogonal to vector Express the answer in component form. Each indicate that mathematical objects are intersecting at right angles. The sum of the forces acting on an object is called the resultant or net force.

In this text, we always work with coordinate systems set up in accordance with the right-hand rule. Some systems do follow a left-hand rule, but the right-hand rule is considered the a night in barcelona yuri on ice standard representation. A bolt is tightened by applying a force of \(6\) N to a 0.15-m wrench (Figure \(\PageIndex\)). The angle between the wrench and the force vector is \(40°\).