Mathematical Physics Vol 1

Chapter 1. Vector algebra

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1.4.2 Multiplication of a vector by a real number (scalar)

Definition Let a be a vector, and α a real number. Then α a ( ≡ a α ) defines a new vector as follows: - if a̸ = 0 and α > 0, the new vector α a has the same direction as vector a , - if a̸ = 0 and α < 0, the new vector α a and the vector a have opposite directions, - the magnitude of α a is equal to | α a | = | α || a | (if a = 0 or α = 0 (or both), then α a = 0 ). It is said that the vector α a is a result of the multiplication of the vector a by the scalar α . We have thus defined the operation of multiplication of a vector by a real number (scalar). The unit vector having the same direction as the vector a will be denoted as e a . Each vector can be represented using the operation of multiplication of a vector by a scalar as a product of its magnitude and its unit vector a = | a | e a . (1.11) For the operation of multiplication of a vector by a scalar the following is true: α a ∈ V , (IIIa) ( α 1 + α 2 ) a = α 1 a + α 2 a (IV) α ( a + b )= α a + α b (V) α 1 ( α 2 a )=( α 1 α 2 ) a , (VI) for each real number α 1 and α 2 and each vector a , b ∈ V . The properties (IV–VI) are known as linearity properties of the set V . 1.4.3 Projection on an axis and on a plane Projection of a point on an axis Consider an axis u determined by a unit vector u , a point A, which does not lie on that axis, and a plane S (Fig. 1.8), which is not parallel to the axis.

Construct a plane S ′ that contains point A and is parallel to plane S . The point A ′ at which the axis u intersects the plane S ′ is the projection of the point A on theaxis u parallel to the plane S . If the plane S is normal to the axis, then the corresponding projection is called normal or orthogonal .

Figure 1.8: Projection of a point on an axis.

Projection of a vector on an axis Let a vector be determined by its start point A and its end point B . By projecting these two points (Fig. 1.9), points A ′ and B ′ are obtained, that is, vector −→ A ′ B ′ .