Question

True or false
The intersection of plane A and plane B is vector PQ.

Answer 1

The intersection of two planes is typically a line, not a single vector. While vector PQ could represent the direction of the intersection line, the statement's accuracy is context-dependent. The other statements delve into various vector properties, some of which are true, such as the applicability of the Pythagorean theorem to vectors at right angles.

Step-by-step explanation:

The intersection of plane A and plane B, according to the question, is vector PQ. Typically, the intersection of two planes in three-dimensional space is a line, not a single vector. Therefore, without additional context, the statement as it stands is false. However, if vector PQ is representative of the direction of the line where the two planes intersect, one could argue that vector PQ does indeed represent that intersection. For clarity and correctness in mathematical terms, the intersection of two non-parallel planes is a line.

The other statements are about different aspects of vectors. Let's go through them one by one:

1. A vector can indeed form the shape of a right angle triangle with its x and y components. This is true because these components act as perpendicular sides of a triangle, with the vector itself being the hypotenuse.
2. Without magnitudes, knowing only the angles of vectors is not enough to calculate the exact angle of the resultant vector. Therefore, the statement 'If only the angles of two vectors are known, we can find the angle of their resultant addition vector' is false.
3. Using the Pythagorean theorem to calculate the length of the resultant vector from two vectors that are at right angles to each other is true. The theorem applies because the vectors form the legs of a right triangle, with the resultant being the hypotenuse.
4. In two-dimensional space, every vector can indeed be expressed as the product of its x and y components, so the statement is true.
5. We cannot find the magnitude and direction of the resultant vector if we only know the angles of two vectors and the magnitude of one. We need the magnitudes of both vectors and the angle between them, thus the statement is false.
6. The direction of the resultant vector does indeed depend on both the magnitude and direction of the added vectors, so this statement is true.
7. If vectors A and B are equal in magnitude and opposite in direction, A - B will have the same direction as vector A because B negates itself, clearing the way for A's direction to prevail. This is true.