the vectors x and y are unit vectors, that makes an angle of 30 degrees with each other. Calculate 3 x-2y.
I guess you could call them
x = <1,0>
y = <√3/2, 1/2>
Then 3x-2y = <3,0> - <√3,1> = <3-√3 , -1>
To calculate 3x - 2y, we need to find the values of x and y.
Given that x and y are unit vectors, we know that their magnitudes are both equal to 1.
Since the vectors x and y make an angle of 30 degrees with each other, we can use the formula to find the dot product of two vectors:
dot product = x1 * y1 + x2 * y2 + x3 * y3
where x1, x2, x3 are the components of vector x, and y1, y2, y3 are the components of vector y.
However, since x and y are unit vectors, we only need to consider their direction.
To find the dot product of unit vectors x and y, we can use the cosine of the angle between them, which is given as 30 degrees.
The dot product formula becomes:
dot product = |x| * |y| * cos(30)
Since |x| = |y| = 1, the formula simplifies to:
dot product = cos(30)
cos(30) = √3/2
Thus, the dot product of x and y is √3/2.
Now, we can calculate 3x - 2y:
3x - 2y = 3 * x - 2 * y = 3 * x - (2 * √3/2) * y
Since x and y are unit vectors, their magnitudes are equal to 1.
Therefore, 3 * x = 3 * 1 * x = 3x
Similarly, (2 * √3/2) * y = √3 * y
So, 3x - 2y = 3x - √3 * y
Finally, we substitute the values obtained for x and y:
3x - 2y = 3 * (√3/2) - √3 * (√3/2)
Simplifying,
3x - 2y = 3√3/2 - 3/2
= (3√3 - 3)/2.
Therefore, the value of 3x - 2y is (3√3 - 3)/2.