Final answer:
It is true that if AB ⟶ and AC ⟶ are opposite rays, they are collinear. The opposite rays extend indefinitely along the same line in opposite directions, making them collinear. Other concepts discussed include the use of the Pythagorean theorem for perpendicular vectors and expressions for vector components in two dimensions.
Step-by-step explanation:
If AB ⟶ and AC ⟶ are opposite rays, then they are collinear. The answer to this is true. Rays AB and AC share the same initial point A and extend in opposite directions along the same line, which makes them collinear by definition. Collinear points are points that lie on the same straight line. Since opposite rays go on indefinitely in opposite directions and cannot bend, they must lie on the same line.
Now, for the other concepts, we must clarify some of the provided options against vector principles:
They point in the same direction. - This is incorrect for opposite rays.
They are perpendicular, forming a 90° angle between each other. - This is incorrect for opposite rays.
They point in opposite directions. - This is correct and true for opposite rays.
They are perpendicular, forming a 270° angle between each other. - This is incorrect and not applicable.
Regarding vectors and the Pythagorean theorem:
We can use the Pythagorean theorem to calculate the length of the resultant vector obtained from the addition of two vectors that are perpendicular (at right angles) to each other. This is true because when two vectors are perpendicular, they form the legs of a right angle triangle and the resultant vector forms the hypotenuse of that triangle.
Vector Components:
A vector can indeed form the shape of a right-angle triangle with its x and y components, which is also true. This is the basis for breaking down a vector into its horizontal (x) and vertical (y) components.
Resultant Vector Calculation:
If only the angles of two vectors are known, we cannot find the angle of their resultant addition vector without additional information such as magnitude; this statement is generally false.
The magnitude and direction of the resultant vector can be found if we know the angles of two vectors and the magnitude of one. This is conditionally true, as with this information, vector addition can be performed using trigonometric methods.
It is also true that every 2-D vector can be expressed as the product of its x and y components, typically represented as Ax = A cos θ and Ay = A sin θ, where θ is the angle the vector makes with the x-axis.