Vectors

We need the resultant vector to be a unit vector along the X-axis. This means the Y and Z components of the resultant vector must be zero. Adding the two vectors: Now, add a vector so that the resultant vector has no Y or Z components and its magnitude is 1. For the X-component, . For the Y-component, . For the Z-component, . Thus, the vector to be added is . Thus, the correct answer is .

We are given the equation of a line . The direction ratios of the line are , which represent the components of the direction vector of the line. These direction ratios also represent the normal vector to the plane because the line is parallel to the plane. Thus, the normal vector of the plane is . The general equation of the plane is: Now, to find , we substitute any point on the line into the equation of the plane. Let's use the point which lies on the line. Substituting into the equation: Thus, the equation of the plane is: Now, comparing this with the options, we find that option (C) is the correct one.

We are given the vectors: We need to find the unit vector in the direction of . First, calculate : Now, combine the vectors: Now, the magnitude of the vector is: Thus, the unit vector is:

We are asked to find the scalar components of a unit vector that is perpendicular to both of the given vectors: - - To find a vector that is perpendicular to both, we take the cross product of and . Step 1: Compute the cross product . The cross product of and is given by the determinant: Expanding the determinant: Now, compute each 2x2 determinant: - For : - For : - For : Thus, the cross product is: Step 2: Normalize the cross product. To find the unit vector, we first compute the magnitude of the cross product: Now, divide the components of the cross product by the magnitude to obtain the unit vector: Thus, the scalar components of the unit vector are: Therefore, the correct answer is option (C)

We are given three vectors: The task is to find the unit vector parallel to the vector , but in the opposite direction.
Step 1: Find
First, add and , then subtract : Now combine the components of :
Step 2: Find the unit vector in the opposite direction
The unit vector in the opposite direction is the negative of the vector , normalized by its magnitude. So we have: Now, find the magnitude of : Thus, the unit vector is: Thus, the unit vector parallel to but in the opposite direction is .

The length of the diagonal of a parallelogram with adjacent sides and is given by the formula: Where: - - Now calculate and : Now find the magnitudes of these vectors: Thus, the length of the diagonals are and .