Vector Algebra

Point: . Line: Point , direction vector .
Distance formula: or use: Answer: 7 units.

For a tetrahedron with coterminous edges , volume is: Proof: The scalar triple product gives the volume of the parallelepiped formed by . A tetrahedron is of this volume.
For given vectors: Answer: Volume = cubic units.

Direction ratios: , . A vector perpendicular to both is given by their cross product: Direction ratios: .

Line 1: Point , direction vector .
Line 2: Point , direction vector .
Shortest distance formula: Answer: .

Lines are coplanar if the vector joining a point on one to a point on the other, and the direction vectors, are coplanar. Points: , . Direction vectors: , .
Coplanarity condition: Lines are coplanar. Plane equation: Normal is . Use point (1, 1, -1): Answer: .

Given , rewrite: If C divides AB in the ratio , the position vector of C is: Equate: Cross-multiply: Equate coefficients of and : - For : .
- For : .
Ratio: .

The vector . The equation is ambiguous, as a scalar times a vector cannot equal a scalar. Assume it means the magnitude of the scaled vector equals 3: Compute the magnitude of : Then: Thus:

We are given the vector equation: Rearranging the equation: This can be interpreted as the vector form of the point dividing the line segment joining and . The formula for the position vector of the point dividing a line segment in the ratio is: Comparing this with our equation, we get: Thus, the ratio in which divides is . So, the ratio is:

The magnitude of the vector is given by: Now, we are given that . So, Thus, the values of are:

Step 1: The direction ratios of the two lines are and . Step 2: To find the direction ratios of the vector perpendicular to both lines, we take the cross product of and . The cross product is given by: Step 3: Compute the determinant: Step 4: Calculate the 2x2 determinants: Step 5: The cross product is: Thus, the direction ratios of the perpendicular vector are:

Step 1: Condition for Vectors to be Collinear
Two vectors and are collinear if there exists a scalar such that: Step 2: Writing the Vectors in Component Form
Given the vectors: Equating the corresponding components: Step 3: Solving for the Scalar
From the equation , solving for : Step 4: Solving for
Substitute into the equation :

Step 1: Vector Equation of a Line
The vector equation of a line passing through and parallel to is: Step 2: Substitute Given Vectors
Step 3: Write the Equation
Expanding: