MCQ related to Dot Product and Cross Product:

1. The dot product of two vectors is always a _______.
a) vector
b) scalar
c) matrix
d) tensor

2. The cross product of two parallel vectors is always _______.
a) a zero vector
b) a unit vector
c) a scalar
d) a vector perpendicular to both vectors

3. The dot product of two perpendicular vectors is always _______.
a) a zero vector
b) a unit vector
c) a scalar
d) a vector perpendicular to both vectors

4. The cross product of two vectors is always _______.
a) a vector
b) a scalar
c) a matrix
d) a tensor

5. The dot product of two vectors can be used to find the _______.
a) length of the vectors
b) angle between the vectors
c) perpendicular distance between the vectors
d) all of the above

6. The cross product of two vectors can be used to find the _______.
a) length of the vectors
b) angle between the vectors
c) perpendicular distance between the vectors
d) area of the parallelogram formed by the vectors

Explanation:

Dot Product: The dot product of two vectors is a scalar quantity obtained by multiplying the corresponding components of the two vectors and adding the products. It is denoted by a dot (·) or by writing the vectors side by side. For example, if a and b are two vectors, then their dot product is given by:

a · b = a1b1 + a2b2 + a3b3

The dot product has various applications in physics and engineering, such as in calculating work done, projection of one vector onto another, and finding the angle between two vectors.

Cross Product: The cross product of two vectors is a vector quantity obtained by multiplying the corresponding components of the two vectors and taking the determinant of the resulting matrix. It is denoted by a cross (×) or by writing the vectors side by side. For example, if a and b are two vectors, then their cross product is given by:

a × b = (a2b3 - a3b2) i + (a3b1 - a1b3) j + (a1b2 - a2b1) k

The cross product has various applications in physics and engineering, such as in calculating torque, finding the normal vector to a surface, and determining the direction of a magnetic field.

In summary, the dot product of two vectors is a scalar quantity, while the cross product of two vectors is a vector quantity. Both have important applications in physics and engineering.