find the angle between vector U=<2,3> and V=<1,-5> answers: 88 degrees, 45 degrees, 135 degrees, or 92 degrees...?
U∘V = |U| |V|cos Ø, where Ø is the angle between
2 - 15 = √13√26cosØ
cosØ = -13/13√2
= -1/√2
Ø = 135°
A point on the terminal side of an angle theta is given. Find the exact value of the six trigonometric functions of theta:
(2,-3)
To find the angle between two vectors, we can use the dot product formula. The dot product of two vectors A = <a1, a2> and B = <b1, b2> is given by:
A · B = a1 * b1 + a2 * b2
In this case, vector U = <2, 3> and vector V = <1, -5>.
Let's calculate the dot product of U and V:
U · V = (2 * 1) + (3 * -5)
= 2 - 15
= -13
The magnitude (length) of a vector A = <a1, a2> is given by:
|A| = sqrt(a1^2 + a2^2)
Let's calculate the magnitudes of U and V:
|U| = sqrt(2^2 + 3^2)
= sqrt(4 + 9)
= sqrt(13)
|V| = sqrt(1^2 + (-5)^2)
= sqrt(1 + 25)
= sqrt(26)
The angle θ between two vectors can be found using the formula:
cos(θ) = (U · V) / (|U| * |V|)
Let's substitute the values:
cos(θ) = -13 / (sqrt(13) * sqrt(26))
To find the angle θ, we can use the inverse cosine function (cos^(-1)):
θ = cos^(-1)(-13 / (sqrt(13) * sqrt(26)))
Using a calculator, we can find the angle θ to be approximately 135 degrees.
Therefore, the correct answer is 135 degrees.