Vectors

We are given the direction ratios of the line and the equation of the plane.
We need to find the angle between the line and the plane.
Step 1: Direction ratios of the line
The direction ratios of the line are given by the coefficients of in the equation of the line. Thus, the direction ratios of the line are .
Step 2: Equation of the plane
The normal vector to the plane is from the equation .
Step 3: Formula for angle between line and plane
The angle between the line and the plane is given by the formula: where is the direction vector of the line and is the normal vector to the plane.
Step 4: Find the dot product and magnitudes
Substitute the values of and to find the dot product: Simplifying the expression: Now, calculate the magnitudes of and : Substitute into the angle formula: Thus, the angle between the line and the plane is .
Step 5: Conclusion
The correct answer is option (a).

We are given a system of equations and asked to find the value of such that the vectors are perpendicular.
Step 1: Interpret the equations as direction ratios
The general form of the direction ratios for a line is , where is the direction vector.
From the given equations: The direction ratios of the line are .

The direction ratios for the second line are .


Step 2: Perpendicular condition
For two lines to be perpendicular, their direction ratios must satisfy the condition: where and are the direction vectors of the two lines. Let the direction vectors be: The dot product is: Simplifying the equation:
Step 3: Solve for
Multiplying the equation by 90k to eliminate the denominators: Thus, .
Step 4: Conclusion
The value of is , corresponding to option (c).

Step 1: Recall the Formulas for Cross and Dot Products The magnitude of the cross product is given by: The magnitude of the dot product is:
Step 2: Relate the Two Magnitudes We are asked to find the relationship between the magnitudes of the cross product and the dot product.
Dividing the cross product magnitude by the dot product magnitude, we get:
Step 3: Conclusion Thus, the correct answer is .

Step 1: Understanding the Problem We are given that , where is a non-zero vector and is a scalar.
The magnitude of a scalar multiple of a vector is given by: We are told that , so:
Step 2: Solve for Solving for , we get: Thus, can be either or , since the absolute value of is .
Step 3: Conclusion Thus, is .