Given that the side length of square ABCD is 2 and E is the midpoint of CD, what is the product of vector BD? And vector AE?

Given that the side length of square ABCD is 2 and E is the midpoint of CD, what is the product of vector BD? And vector AE?

Take B point as the coordinate origin to establish the plane rectangular coordinate system, a (0,2), B (0,0), C (0,2), D (2,2), e (2,1) vector AE (2, - 1), vector BD (2,2) vector AE · vector BD = - 1 × 2 + 2 × 2 = 2

In the square ABCD with side length 1, e and F are the midpoint of BC and DC respectively, then the vector AE · AF=

Square ABCD side length is 1
Let a be the origin, ad be the x-axis and ab be the y-axis rectangular coordinate system
E(1/2,1)F(1,1/2)
Vector AE multiplied by AF = 1 * 1 / 2 + 1 / 2 * 1 = 1

If point E is a moving point in or on the boundary of a square ABCD with side length 2, and F is the midpoint of side BC, then the maximum value of vector af * vector AE is obtained
A 4 B 5 C 6 D 7

Establish the rectangular coordinate system, a as the origin, B (2,0), C (2,2), D (0,2)
Then f (2,1) is the vector AF = (2,1), let AE = (x, y), then the vector af * vector AE = 2x + y
x. Y can't exceed square ABCD, only when x = y = 2, take the maximum value of 6
C 6

It is known that in rectangular ABCD, ab = 2, ad = 1, e and F are the midpoint of BC and CD respectively, then (AE vector + AF vector) × AC vector is equal to?

Because e is the midpoint of BC, AE = 1 / 2 * (AB + AC) = 1 / 2 * (AB + AB + AD) = AB + 1 / 2 * ad, similarly, AF = 1 / 2 * (AC + AD) = 1 / 2 * (AB + AD + ad) = 1 / 2 * AB + ad, so AE + AF = 3 / 2 * (AB + AD) = 3 / 2 * AC, so (AE + AF) * AC = 3 / 2 * AC ^ 2 = 3 / 2 * (AB ^ 2 + ad ^ 2) = 3 / 2 * (4 + 1) = 15 / 2