Product Of Two Vectors

Step 1: Concept
Use the cyclic properties of unit vectors: , , . Also, .
Step 2: Analysis
* . * . * .
Step 3: Conclusion
Sum .
Final Answer: (A)

Step 1: Concept
.
Step 2: Analysis
Since all are unit vectors, . .
Step 3: Conclusion
or .
Final Answer: (C)

Step 1: Concept
This is the standard formula for the Vector Triple Product.
Step 2: Analysis
The vector triple product results in a vector that lies in the plane of and .
Step 3: Conclusion
The expansion is .
Final Answer: (A)

Step 1: Concept
.
Step 2: Analysis
.
Step 3: Conclusion
.
Final Answer: (C)

Step 1: Concept
Projection of on .
Step 2: Analysis
. .
Step 3: Conclusion
Projection .
Final Answer: (A)

Step 1: Concept
The scalar triple product is the determinant of the components of the vectors.
Step 2: Analysis
The vectors are and .
Determinant .
Step 3: Conclusion
.
Final Answer: (A)

Step 1: Concept
The scalar triple product is the determinant of the components of the vectors.
Step 2: Analysis
The vectors are and .
Determinant .
Step 3: Conclusion
.
Final Answer: (A)

Step 1: Concept
Use the cyclic properties of unit vectors: , , . Also, .
Step 2: Analysis
* . * . * .
Step 3: Conclusion
Sum .
Final Answer: (A)

Step 1: Use the given conditions for the vectors.
We are given the following conditions for the vectors : - - - - We also know the relation .

Step 2: Analyze the cross product.
The magnitude of the cross product is given by: Substitute the known values: Thus, .

Step 3: Use the scalar product condition.
We are given that , which implies that the angle between and is the same as the angle between and .

Step 4: Solve for .
We now solve for from the equation . After performing the necessary calculations, we find:

Step 5: Conclusion.
Thus, the value of is , corresponding to option (D).

Step 1: Use the given information.
We are given the conditions: We also know the relation .

Step 2: Use vector cross product properties.
We can use the property of the cross product: Given and , , we substitute: Thus, .

Step 3: Use the scalar product information.
We are also given: This implies that the angle between and is the same as the angle between and .

Step 4: Calculate the value of .
Using the relation , we calculate by solving the cross product relation. The result gives us:

Step 5: Conclusion.
Thus, the value of is , which corresponds to option (C).

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
For :


Since , minimum is negative.
Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}



Continuity at :
Discontinuous at :

From ,
Locus: excluding (intersects )
Step 3: Final Answer

Concept: Angle between vectors:

Step 1: Find .


Step 2: Compute .


Step 3: Find .


Step 4: Find angle.
Conclusion:
Angle =

Concept: A vector perpendicular to both and is parallel to .

Step 1: Compute .

Step 2: Unit vectors. The two unit vectors are:

Concept: The scalar triple product satisfies the identity:

Step 1: Proof outline. Using properties of scalar triple product: The result is the square of the original scalar triple product.

Concept: Volume = absolute value of scalar triple product .

Step 1: Compute .

Step 2: Compute .

Step 3: Volume.

Concept: Scalar triple product is linear in each argument and cyclic.

Step 1: Expand using linearity.

Step 2: Expand cross product.

Step 3: Take dot product with . Many terms cancel due to scalar triple product properties. The result simplifies to:

Concept: Vector identity: .

Step 1: Substitute.

Concept: If two vectors are perpendicular: .

Step 1: Ignore common factor .

Step 2: Compute dot product.

Concept: Cross product is anti-commutative: .

Step 1: Add the vectors.

Step 2: Take magnitude.

Concept: Three vectors are coplanar if their scalar triple product is zero:

Step 1: Write the determinant.

Step 2: Add all rows.

Step 3: Factor from .

Step 4: Perform column operations.

Step 5: Evaluate determinant. The determinant of a triangular matrix is the product of diagonal entries: