Given that a = √ 2, B = 3, the included angle between a and B is 45 °, find the vector a+ λ B and λ When the included angle of a + B is an acute angle, λ Value range of

Given that a = √ 2, B = 3, the included angle between a and B is 45 °, find the vector a+ λ B and λ When the included angle of a + B is an acute angle, λ Value range of

ab=|a||b|cos45°=3
Then vector a+ λ B and λ When the included angle of a + B is an acute angle:
(a+ λ b)( λ a+b)>0
be
λ a ²+ (1+ λ²) ab+ λ b ²> 0
three λ²+ eleven λ+ 3>0
λ> (- 11 + √ 85) / 6, or λ (- 11 + √ 85) / 6, or λ

Given that [vector a, b] satisfies | a | = 3 and | a + B | = | A-B | = 5, find | B|

In this way
a. B are all vectors
（a+b）^2=a^2+b^2+2ab
(a-b)^2=a^2+b^2-2ab
Because | a + B | = | A-B | = 5
So (a + b) ^ 2 = (a-b) ^ 2 = 25
So AB = 0, that is, a and B are perpendicular
So | B | = 4

Vector a + vector b ^ 2 =?

The square of a vector is equal to the square of a vector module
|a+b| ²= (a+b) ²
=a ²+ 2a·b+b ²
=|a| ²+ 2|a|·|b|cos θ+| b| ²
Among them, θ Is the angle between a and B

On a circle with a known radius of 120mm, there is an arc with a length of 144MM. Calculate the chord degree and angle degree of the center angle of the circle with this arc length

Center angle = 1.2
Chord degree = π / 150

If the circumference of a circle with radius 1 is equal to the arc length corresponding to the center angle of 60 °, the radius of the circle where the arc is located is () A.2 B.3 C.4 D.6

D
The circumference of a circle with radius 1 is equal to the arc length corresponding to the center angle of 60 ° = 4 π
Then the arc length corresponding to the center angle of 60 ° and the circumference of the circle is 12 π
Perimeter formula = 2 π R
So radius = 6

The center angle of an arc is 240 °, and the arc length opposite it is equal to the circumference of a circle with a radius of 6cm, then the radius of the circle where the arc is located is equal to________ cm

The arc length is equal to 2 * 6 * 3.14 = 37.68 cm
The radius of the circle where this arc is located is r
2r*3.14*240/360=37.68
4.187r=37.68
R = 9 cm