Vectors

Step 1: We are given the direction ratios of two lines in 3D space, which are for the first line and for the second line.

Step 2: The formula to find the angle between two lines with direction ratios and is:

Step 3: We are given the relation between the direction ratios:

These are two equations in terms of the direction ratios of the lines. We can substitute these relationships into the formula for the cosine of the angle between the lines.

Step 4: After substituting the given conditions and solving for the cosine of the angle, we find that the angle between the lines is:

1. The given equation represents the locus of a point such that the difference in distances from two fixed points and is constant.

2. This is the definition of a hyperbola, where:

3. The distance between the foci is , and the length of the transverse axis is .

4. Therefore, the eccentricity is given by:

Step 1: The unit vector in the XY-plane making an angle of with is given by:

Step 2: The angle between and the vector is . Using the dot product formula:

Step 3: Solving this system of equations gives the components of the unit vector .

1. The coordinates of opposite vertices of a cube are and . The diagonal of the cube is the line connecting these two points.

2. The diagonals of a cube form vectors:

3. The angle between two diagonals is given by:

4. Compute:

5. Substitute:

6. Therefore: