18

ALGEBRA
:: Vectors and a coordinate system, Cartesian vectors
Vectors in a coordinate plane (a two-dimensional system of coordinates), Cartesian vectors
By introducing a coordinate system in a plane with the unit vectors, i and j  (in direction of x and y coordinate axis, respectively) whose tails are in the origin O, then each point of the plane determines a vector rOP.
A directed line segment from the origin to a point P (x, y) in plane is called a radius vector and denoted r.
The radius (or position) vector equals the sum of its vector components, xi and y j  in direction of coordinate axes, that is
Consider a vector a in the plane directed from a point P1(x1, y1) to P2(x2, y2) shown in the right diagram, it also equals to the sum of corresponding vector components axi and ayj, in direction of coordinate axes,
 The diagram shows that are the radius vectors of points P1 and P2,
thus the vector a equals to the difference (joins their heads), that is
It is obvious from the above diagrams that a vector and its components, i.e., its projections in direction of
coordinate axes, form a right triangle, from which, according to Pythagoras’ theorem, we determine the length of the vector,

Example:  Determine a vector a whose tail is at the point P1(-4, 1) and head at the point P2(-1, -3).

 Solution:  Points, P1 and P2 determine radius vectors, therefore,
Vectors in three-dimensional space in terms of Cartesian coordinates

By introducing three mutually perpendicular unit vectors, i, j and k, in direction of coordinate axes of the
three-dimensional coordinate system, called
standard basis vectors, every point P(x, y, z) of the space
determines the radius vector or the position vector,

 or and the length of

Similarly, a vector a in the right diagram, which is directed from a point P1(x1, y1, z1) to a point              P2(x2, y2, z2) in space, equals to sum of its vector components,  axi, ay j, and azk,  in the direction of the coordinate axes, x, y, and z respectively, that is

 The vector a is the difference of the radius vectors,
 thus
so the scalar (numeric) components of the vector a,
Therefore, the length of the vector a,
Angles of vectors in relation to coordinate axes, directional cosines - scalar components of a vector
 The scalar components of a vector and its magnitude form a right triangle in which the hypotenuse equals the magnitude of the vector, then since     b = 90° - a  then  cosb = sina, and components of a vector a,

If a vector a in 3D space forms with the coordinate axes, x, y and z angles, a, b and g respectively, then

 components of the vector are, while for the radius vector of the point P(x, y, z), Thus, for angles, a, b and g hold the expressions, called the directional cosines of the vector a.

The unit vector of the vector a,

This condition must satisfy angles that a vector in three-dimensional space form with the coordinate axes.
 For vectors in a coordinate plane and since  b = 90° - a, then
 cosb = cos(90° - a) = sina,  follows the basic trigonometric identity.
 Similarly, the unit vector of a radius vector in space determined by corresponding directional cosines. Therefore the radius vector forms with coordinate axes angles, As vectors are uniquely determined by its components or coordinates, they are usually denoted using matrix algebra notation, in the coordinate plane, and in the 3-D space
Vectors in a three-dimensional coordinate system, examples
Example:  Determine angles that a radius vector of the point A(3, -2, 5) forms with the coordinate axes.
Solution:  Let calculate the magnitude or length of the radius vector,
 Angles between the radius vector and the coordinate axes are,

Example:  A vector a in a 3D-space, of the length | a | = 4, forms with axes, x and y the same angles,       ab = 60°, find the components (coordinates) of the vector a.

 Solution:  Using relation applying given conditions, The components of a,

Example: Prove that vectors, a = -i + 3 j + kb = 3i - 4 - 2and c = 5i - 10 - 4k are coplanar.

 Solution:  If all three vectors lie on the same plane then there are coefficients, l and m such that, for example c = la + mb, i.e., each of the vectors can be expressed as the linear combination of the remaining two.

Contents B