The vector a = (x, √ 3Y), B = (1,0), and (a + √ 3b) ⊥ (a - √ 3b) are known (1) Find the equation of LOCUS C of point m (x, y) satisfying the above conditions (2) Let the curve C and the straight line y = KX + m (K ≠ 0) intersect at two different points P, Q, and a (0, - 1). When | AP | = | AQ |, the value range of the real number m is obtained

The vector a = (x, √ 3Y), B = (1,0), and (a + √ 3b) ⊥ (a - √ 3b) are known (1) Find the equation of LOCUS C of point m (x, y) satisfying the above conditions (2) Let the curve C and the straight line y = KX + m (K ≠ 0) intersect at two different points P, Q, and a (0, - 1). When | AP | = | AQ |, the value range of the real number m is obtained

∵(a+√3b)⊥(a-√3b).
∴a²-3b²=0
x²+3y²=3
x²/3+y²=1
(2)
LIANLI:
{x²+3y²=3
{y=kx+m
x²+3(kx+m)²=3
(3k²+1)x²+6mkx+3(m²-1)=0
Δx=36m²k²-12(3k²+1)(m²-1)>0
3m²k²>(3k²+1)(m²-1)
3m²k²>(3k²+1)(m²-1)
3k²+1>m²
Let the midpoint of P and Q be m (x0, Y0)
x0=(x1+x2)/2=-3mk/(3k²+1)
y0=kx0+m=m/(3k²+1)
M(-3mk/(3k²+1),m/(3k²+1)),
∵|AP|=|AQ|,
∴AM⊥PQ,
A(0,-1)
K(AM)=(m+3k²+1)/(-3mk)=-1/k
2m=3k²+1>m²
m(m-2)0

Given two points M1 (4, √ 2,1) and M2 (3.0.2), the module, direction cosine and direction angle of vector m1m2 are calculated
Write down the steps, thank you!

M1m2 = (3,0,2) - (4, sqrt (2), 1) = (- 1, - sqrt (2), 1), so: | m1m2 | = sqrt (1 + 2 + 1) = 2 ------ the calculation of modulus can be done directly by coordinate subtraction. This is conducive to the calculation of cosine in three directions: cosa = m1m2 (x) / | m1m2 | = - 1 / 2, so: a = 2 π / 3cosb = m1m2 (y) / | m1m2 | = - sqrt (2) /

Given two points M1 [2,2, radical 2] and M2 [1,3,0], the module, direction cosine and direction angle of vector M1 M2 are calculated

M1 = root sign (square of 2 + square of 2 + square of root sign 2) = follow sign 10m2 = root sign (square of 1 + square of 3 + square of 0) = follow sign 10x, y and Z are the angle between M1 and XYZ axis respectively. The direction cosine of M1 is cosx = 2 times 1 divided by (follow sign 10 times 1) = 5 parts follow sign 10cosy = 2 times 1 divided by (follow sign 10 times 1)