Consider rhombus ABCD
a) Find the resultants of vector AB + vector AD and vector AB - vector AD
b) What will the value of (vector AB + vector AD) dot product (vector AB - vector AD) always be? Explain.
c) Is the value of (vector AB + vector AD) dot product (vector AB - vector AD) the same as your answer to part b)? Explain.
Student
a) To find the resultants of vector AB + vector AD and vector AB - vector AD, we can use the parallelogram law of vector addition.
For vector AB + vector AD, we start by placing the tail of vector AB at the head of vector AD. The resultant vector is the vector that starts at the tail of the first vector and ends at the head of the last vector. Therefore, we draw a vector from point A to point D, which is vector AD.
For vector AB - vector AD, we start by placing the tail of vector AD at the head of vector AB. The resultant vector is the vector that starts at the tail of the first vector and ends at the head of the last vector. Therefore, we draw a vector from point A to point B, which is vector AB.
b) The dot product of two vectors, let's call them vector U and vector V, is defined as: U · V = |U| |V| cosθ, where |U| and |V| are the magnitudes of the vectors, and θ is the angle between them.
In the case of vector AB + vector AD and vector AB - vector AD, the angle between them is 180 degrees, as they are opposite vectors.
Since cos(180 degrees) = -1, the dot product of (vector AB + vector AD) and (vector AB - vector AD) will always be:
(vector AB + vector AD) · (vector AB - vector AD) = |vector AB + vector AD| |vector AB - vector AD| cos(180 degrees)
= |vector AB + vector AD| |vector AB - vector AD| (-1)
= -|vector AB + vector AD| |vector AB - vector AD|
c) Yes, the value of (vector AB + vector AD) · (vector AB - vector AD) is indeed the same as the value calculated in part b) because the dot product does not depend on the specific vectors used, but rather the properties of the dot product itself. In part b), we used the definition of the dot product and the fact that the angle between (vector AB + vector AD) and (vector AB - vector AD) is 180 degrees to find its value. As a result, the magnitude of the dot product is always equal to -|vector AB + vector AD| |vector AB - vector AD|.