We discuss problems, which highlight certain aspects of the vector product. For this reason, questions are categorized in terms of the characterizing features of the subject matter :

- Condition of parallel vectors
- Unit vector of cross product
- Nature of vector product
- Evaluation of vector product
- Area of parallelogram

** Condition of parallel vectors
**

#### Example 1

Problem : Determine whether vectors 2i – j + 2k and 3i – 3j + 6k are parallel to each other?

Solution : If the two vectors are parallel, then ratios of corresponding components of vectors in three coordinate directions are equal. Here,

The ratios are, therefore, not equal. Hence, given vectors are not parallel to each other.

** Unit vector of cross product
**

#### Example 2

Problem : Find unit vector in the direction perpendicular to vectors i + j – 2k and 2i –j + 3k.

Solution : We know that cross product of two vectors is perpendicular to each of vectors. Thus, unit vector in the direction of cross product is perpendicular to the given vectors. Now, unit vector of cross product is given by :

Vector product |
---|

Here,

** Nature of vector product
**

#### Example 3

Problem : Verify vector equality B = C, if AxB = AxC.

Solution : Let

It is clear that B=C is true only when

** Evaluation of vector product
**

#### Example 4

Problem : If a.b = |axb| for unit vectors a and b, then find the angle between unit vectors.

Solution : According to question,

#### Example 5

Problem : Prove that :

Solution : Expanding LHS, we have :

** Area of parallelogram
**

#### Example 6

Problem : The diagonals of a parallelogram are represented by vectors 3i+j+k and i-j-k. Find the area of parallelogram.

Solution : The area of parallelogram whose sides are formed by vectors a and b, is given by :

However, we are given in question vectors representing diagonals – not the sides. But, we know that the diagonals are sum and difference of vectors representing sides of a parallelogram. It means that :

Diagonals of a parallelogram |
---|

and

Now the vector product of vectors representing diagonals is :

Using anti-commutative property of vector product,

Thus,

The volume of the parallelogram is :