Let vector a = (cos23, cos67), vector b = (cos68, cos22), vector u = vector a + T, vector B, and find the minimum value of the module of U

Let vector a = (cos23, cos67), vector b = (cos68, cos22), vector u = vector a + T, vector B, and find the minimum value of the module of U

Vector a = (cos23, cos67), vector b = (cos68, cos22)
The vector a | = √ (COS & # 178; 23 + cos & # 178; 67) = √ (COS & # 178; 23 + Sin & # 178; 23) = 1
|Vector B | = √ (COS & # 178; 68 + cos & # 178; 22) = √ (Sin & # 178; 22 + cos & # 178; 22) = 1
Vector A. vector b = cos23cos68 + cos67cos22
=sin22cos23+cos22sin23
=sin45
=√2/2
| vector u | & # 178; = (a + TB) & # 178;
=a²+t²b²+2t a.b
=t²+√2t+1
=(t+√2/2)²+1/2
When t = - √ 2 / 2, the | vector u | & # 178 has a minimum value of 1 / 2,
The minimum value of | u | is √ 2 / 2

Simplification of cos15 ° cos45 ° - cos75 ° sin45 °
Can cos 15 ° turn into sin 75?

cos15°cos45°-cos75°sin45°
=sin75°cos45°-cos75°sin45°
=sin(75°-45°）
=1/2
cos15°=cos（90°-75°）=sin75°
I hope the answer can help you,

Let a = (cos23, cos67), B = (cos68, cos22), u = a + T, B (t belongs to R)
1) Find A.B;
2) Finding the minimum of the module of U
"58.241.39." * "your first question is wrong too..."

A = (cos23, sin23), B = (cos68, sin68) | a | = | B | = 11. A * b = cos23cos68 + sin23sin68 = sin (23-68) = cod (- 45) = cos45 = √ 2 / 22. U = a + TB | u | ^ 2 = (a + TB) ^ 2 = a ^ 2 + 2tab + T ^ 2 = T ^ 2 + √ 2T + 1 = (T + √ 2 / 2) ^ 2 + 1 / 2