Vectors in a Plane and  Space
     Vector product or cross product
      The mixed product or the scalar triple product
      The vector product and the mixed product use, examples
The vector product and the mixed product use examples
Example:   Given are vectors, a = i - 2 k and  b = - i + 3 j + k, determine the vector  c a ´ b and the area of a parallelogram formed by vectors, a and b.
The area of the parallelogram,  
Example:   Vertices of a triangle are, A(-1, 0, 1), B(3, -2, 0) and C(4, 1, -2), find the length of the altitude hb.
Solution:  Using the area of a triangle,
as the area of the triangle where
then, the area of the triangle,  
and the length of the side b,  
thus, the length of the altitude,
Example:  Find the angles that the unit vector, which is orthogonal to the plane formed by vectors, 
a = -i + 2 j + k  and  b = 3i - 2 j + 4k, makes with the coordinate axes.
Solution:  The unit vector which is orthogonal to the plane, formed by the vectors, a and b, is the direction vector of a vector c, such that
Angles, which the unit vector c°  forms with coordinate axes, we find by using the formula,
Example:  Given are vectors, a = -2i - 3 jb = -i - 2 j + 3k and c = -i + 2 j + k, find the projection (the scalar component) of the vector a onto vector d = b ´ c.
Solution:  Let first find the vector d,
Example:  Given are vectors, a = -3i + 4 j - l kb = 2i -  j + k  and  c = i - 4 j -3l k, determine the 
l such that vectors to be coplanar.
Solution:  Vectors lie on the same plane if their scalar triple product is zero, i.e., V = 0, therefore vectors’
coordinates must satisfy the condition,
Example:  Examine if vectors, a = 4i + 2 + kb = 3i + 3 j  - 2k  and  c = - 5i -  j - 4k, are coplanar and if so, prove their linear dependence.
Solution:  Check for coplanarity,
Since vectors are coplanar each of them can be represented as linear combination of other two.
Example:  Prove that points, A(-2, 1, 0), B(3, -2, -1), C(1, -1, 2) and D(-3, 2, -5), all lie on the same plane.
Solution:  If all given points belong to the same plane then vectors, AB, AC and AD are coplanar, therefore the scalar triple product (AB ´ AC) · AD = 0.
thus, all given points lie on the same plane.
Vectors in 2D and 3D Contents
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