The vector method in solid geometry to prove the formula of parallel and vertical

The vector method in solid geometry to prove the formula of parallel and vertical

Vector a = (x1, Y1, z1) B = (X2, Y2, Z2)
A / / B then X1 / x2 = Y1 = y2 = Z1 / Z2
A ⊥ B, then x1x2 + y1y2 + z1z2 = 0

The scalar product of vector * 2
Given that the angle between a and B is 30 degrees, the module of a = root 3, and the module of B = 1, the cosine of the angle between a + B and A-B is calculated
In the parallelogram ABCD, if the absolute value of AB module is 4, the absolute value of ad module is 3, and the angle DAB is 60, then the vector AB * vector CD =? Vector ad * vector DC =?

ab=3/2
(a+b)^2=3+1+3=7
(a-b)^2=3+1-3=1
Cos θ = (3-1) / (root 7 * root 1) = 2 / 7 root 7
Vector AB * vector CD = - AB ^ 2 = - 4 ^ 2 = - 16
Vector ad * vector DC = vector ad * vector AB = 4 * 3 * cos60 = 6
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Is the square of vector a a vector or a distance?

A ^ 2 = a * a = | a | * | a | * cos0 ° = | a | ^ 2, which obviously refers to the distance
In fact, you can also think that a ^ 2 can be regarded as a * B, but the vector b = a

Square of vector
I is parallel to the X axis
J is parallel to the Y axis
（3i+5j)^2=?

(3i+5j)^2=9i^2+15i*j+25j^2=9+0+25=34