§ 6 projection of vectors on the axis (projection)

§ 6 projection of vectors on the axis (projection)

 

 

IAngle between two vectors of space:

There are two vectors that are handed over to the point (if they are not intersecting, one of the vectors can be translated to make them cross), and one of the vectors is rotated inside the plane determined by the two vectors, so that the square direction of it overlaps with the square direction of another constant, so that the obtained rotation angle (Limitation) is calledAngle,.

(Fig. 1.17)

If parallel, when they point to the same time, the angle between them is specified; when they point to the opposite, the angle between them is specified.

Similarly, the angle between the vector and the number axis can be specified.

Move the vector in parallel to the intersection of the number axis, and then rotate the vector around the intersection point in the plane determined by the vector and number axis, so that the square direction of the vector overlaps with the square direction of the number axis, the obtained rotation angle is called the angle between the vector and the number axis..

(Fig. 1.18)

Projection of two spatial points on the Axis:

If a known point and an axis are used as the vertical plane of the axis, the intersection of the plane and the axis is called the projection of the point on the axis.

(Fig. 1.19)

Projection of three vectors on the Axis:

Definition 1If the projection of the start point and end point of a vector on the axis is respectively, then the value of the directed line segment on the axis is called the projection of the vector on the axis.

(Fig. 1.20)

Here, the value is such a number:

(1), that is, the absolute value of a number is equal to the modulus of a vector.

(2) When the direction is the same as that of the axis, when the direction is the opposite to that of the axis ,.

ThuProjection Theorem:

Theorem 1The projection of a vector on the axis is equal to the modulus of the vector multiplied by the cosine of the angle between the axis and the vector. That is

, (1.6-1)

(Fig. 1.21)

CertificateIf the starting point of an over-vector leads the axis, and the axis is parallel to the axis and has the same positive direction, the angle between the non-axis and the vector is equal to the angle between the axis and the vector, and there are

Hence

We can see from the above formula that the projection of a vector on the axis is a numerical value, not a vector.

When the non-zero vector and the projection axis form an acute angle, the projection of the vector is positive.

Theorem 2For any vector

. (1.6-2)

CertificateIf the projection is respectively on the axis, then obviously,

Because

So,

That is.

Likewise, the following theorem can be proved:

Theorem 3For any vector and any real number

. (1.6-3)