Vector Algebra

Step 1: Given,

So, is perpendicular to .

Step 2: Given,

So, is parallel to .

Step 3:
So we have:
- is perpendicular to
- is also parallel to

The only vector that can be both perpendicular and parallel to the same non-zero vector is the zero vector.

Therefore,

To find , we use the relationship: Given that and , we calculate :
Simplifying:

The area of parallelogram ABCD is given by the magnitude of the cross product of vectors and :

The cross product of and is computed as follows:

Expanding the determinant:

Calculating each 2x2 determinant: Thus,

The magnitude of the cross product is:

Answer: The magnitude of the cross product is , so the area of parallelogram ABCD is . \bigskip

The vector from to is given by: where are the coordinates of point , and are the coordinates of point .

Substituting the given coordinates:

Simplify each term:

Thus:

Hence, the vector is , and the correct answer is (D).

The dot product of two unit vectors is given by:

Using the Pythagorean identity:

Substitute : Simplify:

Thus:

Final Answer: , (C).

The area of a parallelogram formed by vectors and is given by: We are given: Recall that: Now, compute using the identity: Now compute the area:

Final Answer: