Geometry

We are given the equation: Simplifying both sides: Thus, the value of is:

For two straight lines to be perpendicular, the dot product of their direction ratios must be zero. Let the direction ratios of the first line be and the direction ratios of the second line be . The dot product of the two direction ratios is given by: This condition must hold for the lines to be perpendicular. Thus, the correct condition for the lines to be perpendicular is: Hence, the correct answer is:

The direction cosines of a line are given by the formula: where are the direction ratios of the line. Given the direction ratios , we first calculate the magnitude of the direction ratios: Now, we calculate the direction cosines: Thus, the correct answer is:

The equation of a plane is given by the general form: where are the coefficients of the plane and is a constant. The key point here is that parallel planes have the same normal vector, which means their coefficients are identical, but their constant term is different. The given plane is: A plane parallel to this one will have the same coefficients of , but the constant term will be different. Therefore, the equation of a parallel plane will have the form: where is any constant different from 11. Thus, the correct answer is:

The angle between two planes is given by the formula: where and are the normal vectors of the two planes. For the first plane , the normal vector . For the second plane , the normal vector . Now, compute the dot product of the two normal vectors: Next, compute the magnitudes of the normal vectors: Now, substitute these values into the formula for : Thus, the angle between the planes is: Hence, the correct answer is:

The distance of a point from a plane is given by: Rewrite the plane equation as: Here, and the point is . Substitute into the distance formula: Rationalizing the denominator: Since none of the options exactly matches this simplified form, the closest fractional form is: The correct option is:

Two planes are perpendicular if and only if their normal vectors are perpendicular. Let the normal vectors be: The condition for perpendicularity is: Calculate the dot product: Thus, the value of is:

For two lines to be parallel, their direction ratios must be proportional. The direction ratios of the first line are . The direction ratios of the second line are . Therefore, which implies Thus, the correct option is:

For the line to be parallel to the plane, the direction ratios of the line must be perpendicular to the normal vector of the plane. The direction ratios of the line are , and the normal vector to the plane is . The condition for perpendicularity is given by the dot product of the direction ratios of the line and the normal vector of the plane being zero: Thus, the correct option is:

The equation of a plane parallel to another plane is obtained by keeping the same direction ratios for the normal vector, but changing the constant. Here, the normal vector of the given plane is . For a plane parallel to this one, we keep the same direction ratios but change the constant to get .

The equation of the line is given as: Let the common ratio be . Then, we can write the parametric equations for , , and as: Now, we substitute the coordinates into these equations to verify the point: Substituting into the equation for : Substituting into the equation for : Thus, the line passes through the point (12, 13, 14). Hence, the correct answer is option (C).

The direction ratios of two parallel lines are proportional, i.e., the direction ratios of both lines must be equal up to a constant multiple. Let's use the following direction ratios for the first line : , and for the second line , the direction ratios are . For parallel lines, we have: From the equation , we can write: Now, from , we can write: Substitute into the equation for : Thus, the direction ratios for the second line are . We can now substitute these into the equation of the line to find . Since the direction ratios are proportional to the first line, the value of should match the scaling factor. Hence, solving gives: Thus, the correct answer is option (B).

For two parallel lines, the direction ratios are proportional. This means the corresponding direction ratios of both lines should be proportional to each other. If we have direction ratios for the first line and for the second line, we have: From this, we can find that , which means the correct relation between the direction ratios is .

For two mutually perpendicular lines, the dot product of their direction ratios must be zero. Let's take the direction ratios of the first line as and the second line as . The condition for mutual perpendicularity is: Simplifying: Thus, the correct answer is option (B).