Let u = [2,−3, 1] and v = [0, 1, 4]. Calculate the vector u − v. Place u and v tail-to-tail to form two sides of a triangle. With regard to this triangle, what does u−v represent?
Calculate the number u•u and discuss its relevance to the diagram you have drawn. Do the same for the number (u − v) • (u − v).
To calculate the vector u - v, we subtract the corresponding components of u and v:
u - v = [2, -3, 1] - [0, 1, 4] = [2-0, -3-1, 1-4] = [2, -4, -3].
Now, when we place vectors u and v tail-to-tail to form two sides of a triangle, the vector u - v represents the third side of the triangle, from the tail of v to the head of u. This is known as the difference vector between u and v.
To calculate the dot product u • u, we take the dot product of u with itself:
u • u = (2)(2) + (-3)(-3) + (1)(1) = 4 + 9 + 1 = 14.
In this context, the dot product u • u represents the square of the magnitude (length) of vector u. It tells us how long the vector u is.
Similarly, to calculate (u - v) • (u - v), we take the dot product of the vector u - v with itself:
(u - v) • (u - v) = (2)(2) + (-4)(-4) + (-3)(-3) = 4 + 16 + 9 = 29.
In the diagram we have drawn, the dot product (u - v) • (u - v) represents the square of the magnitude (length) of the vector u - v. It tells us how long the difference vector u - v is and relates to the size of this third side of the triangle formed by u and v.