If 'a' and 'b' are unit vectors that make an angle of 60 degrees with each other, calculate
l 3a - 5b l and l 8a + 3b l
the 'a' and 'b' have a carat of top of them
Let a=<1,0>
Then b=<1/2,√3/2>
Now you just add the components.
For example,
2a+3b = <2,0>+<3/2,3√3/2>
and |2a+3b|=√((2+3/2)^2+(0+3√3/2)^2) = √(49/4 + 27/4) = √76/2 = √19
To calculate the lengths of the given vectors, we can use the formula for calculating the magnitude of a vector:
Magnitude of vector = √((x^2) + (y^2) + (z^2))
In this case, since we have unit vectors, we can simplify the calculation as follows:
1. For vector 3a - 5b:
x = 3 * magnitude of vector a * cos(angle between a and b)
= 3 * 1 * cos(60 degrees) = 3 * 1 * (1/2) = 3/2
y = 3 * magnitude of vector a * sin(angle between a and b)
= 3 * 1 * sin(60 degrees) = 3 * 1 * sqrt(3)/2 = (3 * sqrt(3))/2
z = 0 (since it's a unit vector)
Hence, the vector 3a - 5b can be written as (3/2)i + (3 * sqrt(3))/2j + 0k
The magnitude of the vector = √((3/2)^2 + ((3 * sqrt(3))/2)^2 + 0^2)
= √(9/4 + (27/4) + 0)
= √(36/4)
= √9
= 3
Therefore, the length of the vector 3a - 5b is 3.
2. For vector 8a + 3b:
Similar calculations can be done:
x = 8 * magnitude of vector a * cos(angle between a and b)
= 8 * 1 * cos(60 degrees) = 8 * 1 * (1/2) = 4
y = 8 * magnitude of vector a * sin(angle between a and b)
= 8 * 1 * sin(60 degrees) = 8 * 1 * sqrt(3)/2 = 4 * sqrt(3)
z = 0 (since it's a unit vector)
Hence, the vector 8a + 3b can be written as 4i + 4 * sqrt(3)j + 0k
The magnitude of the vector = √(4^2 + (4 * sqrt(3))^2 + 0^2)
= √(16 + 48 + 0)
= √64
= 8
Therefore, the length of the vector 8a + 3b is 8.
To summarize:
- The length of the vector 3a - 5b is 3.
- The length of the vector 8a + 3b is 8.