Vectors in a Plane and  Space
Vectors and a coordinate system, Cartesian vectors
Projection of a vector in the direction of another vector, the scalar and vector components
The scalar component
The vector component
Scalar product of vectors examples
Projection of a vector in the direction of another vector, the scalar and vector components
The scalar component
The length of projection of a in the direction of b or the scalar component ab, from the diagram,
Thus, the scalar component of a vector a in the direction of a vector b equals the scalar product of the vector a and the unit vector b0 of the vector b.
The vector component
By multiplying the scalar component ab, of a vector a in the direction of b, by the unit vector b0 of the vector b, obtained is the vector component of  a in the direction of b.
 or
Scalar product of vectors examples
Example:   Applying the scalar product, prove Thales’ theorem which states that an angle inscribed in a
semicircle is a right angle.
 Solution:  According to diagram in the right figure since square of a vector equal to square of its length,
 thus,  a and b are orthogonal vectors as .
Example:   Prove the law of cosines used in the trigonometry of oblique triangles.
 Solution:  Assuming the directions of vectors as in the right diagram Using the scalar product and substituting square of vectors by square of their lengths, obtained is a2 = b2 + c2 - 2bc · cosa  - the law of cosines.
In the same way,
Example:   Determine a parameter l so the given vectors, a = -2i + l j - 4k  and  b = i - 6 j + 3k  to be perpendicular.
Solution:  Two vectors are perpendicular if their scalar product is zero, therefore
Example:   Find the scalar product of vectors, a = -3mn and  b = 2m - 4n if  | m | = 3 and  | n | = 5 , and the angle between vectors, m and n is 60°.
 Solution:
Example:   Given are vertices, A(-2, 0, 5), B(-3, -3, 2), C(1, -2, 0) and D(2, 1, 3), of a parallelogram,
find the angle subtended by its diagonals as is shown in the diagram below.
 Solution:
Example:   Given are points, A(-2, -3, 1), B(3, -1, -4), C(0, 2, -1) and D(-3, 0, 2), determine the scalar and vector components of the vector AC onto vector BD.
Solution:  The scalar component of the vector AC onto vector BD,
The vector component of the vector AC in the direction of the vector BD equals the product of the scalar component ACBD and the unit vector BD° that is
Vectors in 2D and 3D Contents