Mathematical Physics Vol 1

1.4 Operations on vectors

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Figure 1.9: Projection of a vector on an axis.

The projection of a vector on an axis is a scalar called the algebraic value of the projection or shortly projection . Thus, the projection of a vector on an axis is a scalar. The algebraic value of the projection of the vector −→ AB is denoted by A ′ B ′ , and defined by:

   −

+ −→ A ′ B ′ ,

if the vector −→ A ′ B ′ has the same direction as the axis u ,

A ′ B ′ =

−→ A ′ B ′ ,

if the vector −→ A ′ B ′ and the axis u have opposite directions .

If the angle between the vector −→ AB and the vector u of the axis u is denoted by α , then A ′ B ′ = proj u −→ AB = | −→ AB | cos α .

R Note that the following proposition holds: the projection (algebraic value of the projection) of a sum of vectors on an arbitrary axis, is equal to the sum of the projections of these vectors, parts of the sum, on that axis.

Projection of a point and a vector on a plane In order to project a point ( A ) on a plane ( S ), it is necessary to first select a straight line ( p )with respect to which the point we will be projected. The intersection ( A ′ ) of the plane ( S ) and the line ( p 1 ), ( p ∥ p 1 ), to which point ( A ) belongs, is called the projection of point A onaplane ( S ) in the direction of straight line ( p ) (Fig. 1.10). If the line ( p ) is normal to the plane ( S ), then the corresponding projection is called normal (orthogonal). The projection of a vector on a plane is obtained by projecting its start and end points (Fig. 1.10).

Figure 1.10: Projection of a point and a vector on a plane.

Thus, the projection of a vector on a plane is a vector.