Vector Algebra

The correct answer is A:
Given that: are the magnitudes of vectors;







and
As question asked for the order of arrangement of so it will be;

and
Let


and


Area of parallelogram

( and are diagonals)





sq units

Since, , therefore

Also,

Hence,

Step 1: Writing the system of equations as a matrix.
The system of equations can be written in matrix form as follows:
For the system to have a non-zero solution, the determinant of the coefficient matrix must be zero.
Step 2: Finding the determinant.
The determinant of the matrix is: Expanding the determinant and solving gives the condition for .
Step 3: Solving for .
After solving the determinant equation, we find that satisfies the condition for a non-zero solution.
Conclusion.
The correct answer is (3) , as results in a non-zero solution.

and are reciprocal vectors of and respectively.

By simplifying the given expression using algebraic manipulation, we can solve for as .

By applying vector and reciprocal vector properties, we find that is the correct expression.

Step 1: The cross product of and gives the vector perpendicular to both and . Use the given cross product result to determine .
Step 2: After solving for , we find .

Final Answer:

Step 1: The unit vector perpendicular to a plane is given by the cross product of two vectors lying on the plane.
Step 2: Using the given points, we form two vectors and compute their cross product. After normalizing the result, we get .

Final Answer:

Step 1: The projection of a vector on the xy-plane is given by the vector .
Step 2: The length of the projection is the magnitude of :

Final Answer:

Step 1: To solve for , calculate the determinant of the coefficient matrix. For non-trivial solutions, the determinant should be zero.
Step 2: After calculating, we find that .

Final Answer:

Since,

The given line is

This shows that the given line passes through the point and has direction ratios Thus, given line passes through the point having position vector and is parallel to the vector
. So, its vector equation is

Hence,

The correct answer is A:
Given that;
Position vector of A & B are;
&
respectively


Now let internal bisector OP divides AB with ratio
Position vector of D is