Question

How to tell whether the given points are parallel, perpendicular, or neither and explain your answer with why.​

Answer 1

To determine if given points are parallel or perpendicular, vectors connecting the points are considered. Parallel vectors have the same or opposite direction but the same slope, and perpendicular vectors form a 90-degree angle. We use vector component equations along x and y axes to find their directions.

Step-by-step explanation:

To determine whether given points are parallel, perpendicular, or neither, we must first understand that this question likely refers to the direction of vectors or lines that pass through these points rather than the points themselves. Points do not have direction, but lines or vectors that connect two points do.

Step 1. Identify the x- and y-axes to be used as reference. Then, we need to find the components of each vector along these axes using the equations Ax = A cos θ and Ay = A sin θ, where Ax and Ay are the components along the x-axis and y-axis respectively, A is the magnitude of the vector, and θ is the angle the vector makes with the x-axis.

If vectors are parallel, they will have the same or opposite direction but the same slope when placed in the same coordinate system. However, if vectors are perpendicular to each other, they will form a 90-degree angle, which implies that the product of their slopes will be -1. If the vectors are neither parallel nor perpendicular, they will not satisfy either condition.

To elaborate on the options provided:

• a. All three are parallel to each other and are along the x-axis: This suggests that all vectors are horizontal and share the same slope, which is zero.
• b. All three are mutually perpendicular to each other: This means every pair of vectors forms a 90-degree angle with one another.
• c. They point in opposite directions: This implies that the vectors are parallel but have opposite signs for their slopes.
• d. They are perpendicular, forming a 270° angle between each other: Here, perpendicularity is maintained despite the unusual angle measurement, as 270° is effectively the same as 90° with regard to perpendicularity in a plane.