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By the end of this section, you will be able to:
  • Explain the difference between the scalar product and the vector product of two vectors.
  • Determine the scalar product of two vectors.
  • Determine the vector product of two vectors.
  • Describe how the products of vectors are used in physics.

A vector can be multiplied by another vector but may not be divided by another vector. There are two kinds of products of vectors used broadly in physics and engineering. One kind of multiplication is a scalar multiplication of two vectors . Taking a scalar product of two vectors results in a number (a scalar), as its name indicates. Scalar products are used to define work and energy relations. For example, the work that a force (a vector) performs on an object while causing its displacement (a vector) is defined as a scalar product of the force vector with the displacement vector. A quite different kind of multiplication is a vector multiplication of vectors . Taking a vector product of two vectors returns as a result a vector, as its name suggests. Vector products are used to define other derived vector quantities. For example, in describing rotations, a vector quantity called torque is defined as a vector product of an applied force (a vector) and its distance from pivot to force (a vector). It is important to distinguish between these two kinds of vector multiplications because the scalar product is a scalar quantity and a vector product is a vector quantity.

The scalar product of two vectors (the dot product)

Scalar multiplication of two vectors yields a scalar product.

Scalar product (dot product)

The scalar product     A · B of two vectors A and B is a number defined by the equation

A · B = A B cos φ ,

where φ is the angle between the vectors (shown in [link] ). The scalar product is also called the dot product    because of the dot notation that indicates it.

In the definition of the dot product, the direction of angle φ does not matter, and φ can be measured from either of the two vectors to the other because cos φ = cos ( φ ) = cos ( 2 π φ ) . The dot product is a negative number when 90 ° < φ 180 ° and is a positive number when 0 ° φ < 90 ° . Moreover, the dot product of two parallel vectors is A · B = A B cos 0 ° = A B , and the dot product of two antiparallel vectors is A · B = A B cos 180 ° = A B . The scalar product of two orthogonal vectors vanishes: A · B = A B cos 90 ° = 0 . The scalar product of a vector with itself is the square of its magnitude:

A 2 A · A = A A cos 0 ° = A 2 .
Figure a: vectors A and B are shown tail to tail. A is longer than B. The angle between them is phi. Figure b: Vector B is extended using a dashed line and another dashed line is drawn from the head of A to the extension of B, perpendicular to B. A sub perpendicular is equal to A magnitude times cosine phi and is the distance from the vertex where the tails of A and B meet to the location where the perpendicular from A to B meets the extension of B. Figure c: A dashed line is drawn from the head of B to A, perpendicular to A. The distance from the tails of A and B to where the dashed line meets B is B sub perpendicular and is equal to magnitude B times cosine phi.
The scalar product of two vectors. (a) The angle between the two vectors. (b) The orthogonal projection A of vector A onto the direction of vector B . (c) The orthogonal projection B of vector B onto the direction of vector A .

The scalar product

For the vectors shown in [link] , find the scalar product A · F .

Strategy

From [link] , the magnitudes of vectors A and F are A = 10.0 and F = 20.0. Angle θ , between them, is the difference: θ = φ α = 110 ° 35 ° = 75 ° . Substituting these values into [link] gives the scalar product.

Solution

A straightforward calculation gives us

A · F = A F cos θ = ( 10.0 ) ( 20.0 ) cos 75 ° = 51.76 .
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Check Your Understanding For the vectors given in [link] , find the scalar products A · B and F · C .

A · B = −57.3 , F · C = 27.8

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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