Use the law of cosines to find the angle of Q between the given vectors.
v=3i + j w=2i - j
\vec b\cdot \vec c = \Vert \vec b\Vert\Vert\vec c\Vert\cos \theta
The law of cosines in vector form:
b dot c= abs b * abs c *CosA
(3i+j)dot(2i-j)=sqrt(10)*sqrt(5) CosA
CosA=5/5sqrt2= .707
solve for A
To find the angle between two vectors using the law of cosines, we first need to find the magnitudes of the vectors and the dot product of the vectors.
Step 1: Find the magnitudes of the vectors.
The magnitude of a vector is given as the square root of the sum of the squares of its components.
The magnitude of vector v is given by:
|v| = sqrt((3^2) + (1^2)) = sqrt(9 + 1) = sqrt(10)
The magnitude of vector w is given by:
|w| = sqrt((2^2) + (-1^2)) = sqrt(4 + 1) = sqrt(5)
Step 2: Find the dot product of the vectors.
The dot product of two vectors v and w is given by the sum of the products of their corresponding components.
The dot product of vectors v and w is given by:
v · w = (3 * 2) + (1 * -1) = 6 - 1 = 5
Step 3: Apply the law of cosines.
The law of cosines states that for an angle Q between two vectors v and w, the following relationship holds:
cos(Q) = (v · w) / (|v| * |w|)
In this case, we can substitute the values we found earlier:
cos(Q) = 5 / (sqrt(10) * sqrt(5))
Step 4: Calculate the angle Q.
To find the angle Q, we can take the inverse cosine (or arccosine) of both sides:
Q = cos^(-1)(5 / (sqrt(10) * sqrt(5)))
Using a calculator, we can evaluate this expression to find the value of Q.