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*
How do i answer this without using components?
I guess you could use the law of cosines.
Draw a diagram and you will see that
|3a-5b|^2 = 3^2+5^2-2*3*5*cos120°
do the same for the other
To answer this question without using vector components, we can employ the dot product formula and trigonometry.
The dot product between two vectors, denoted by "•" or "( )", is defined as the product of their magnitudes and the cosine of the angle between them. The formula for the dot product is:
a • b = |a| * |b| * cosθ
First, let's calculate the dot product of 'a' and 'b' using the given information. Since 'a' and 'b' are unit vectors, their magnitudes are both 1. The angle between them is given as 60 degrees.
a • b = |a| * |b| * cosθ
= 1 * 1 * cos(60°)
To evaluate cos(60°), we can use the value of the cosine function for the angle 60 degrees, which is 1/2.
a • b = (1)(1)(1/2)
= 1/2
Now, let's calculate the magnitude of '3a - 5b'. The magnitude of a vector can be found using the formula:
|3a - 5b| = √[(3a - 5b) • (3a - 5b)]
Using the dot product, we can simplify this:
|3a - 5b| = √[(3a - 5b) • (3a - 5b)]
= √[(3a • 3a) + (3a • -5b) + (-5b • 3a) + (-5b • -5b)]
Since we already know that a • b = 1/2, we can substitute this value in the formula:
|3a - 5b| = √[(3a • 3a) + (3a • -5b) + (-5b • 3a) + (-5b • -5b)]
= √[(9)(a • a) + (-15)(a • b) + (-15)(b • a) + (25)(b • b)]
Since a • a and b • b are the magnitudes of 'a' and 'b', respectively, and both 'a' and 'b' are unit vectors with magnitude 1, we have:
|3a - 5b| = √[(9)(1) + (-15)(1/2) + (-15)(1/2) + (25)(1)]
= √[9 - 7.5 - 7.5 + 25]
= √[19]
Therefore, |3a - 5b| = √[19] or approximately 4.357.
Similarly, let's calculate the magnitude of '8a + 3b':
|8a + 3b| = √[(8a + 3b) • (8a + 3b)]
= √[(8a • 8a) + (8a • 3b) + (3b • 8a) + (3b • 3b)]
Since we already know that a • b = 1/2, we can substitute this value in the formula:
|8a + 3b| = √[(8a • 8a) + (8a • 3b) + (3b • 8a) + (3b • 3b)]
= √[(64)(a • a) + (24)(a • b) + (24)(b • a) + (9)(b • b)]
With the same reasoning as before, we have:
|8a + 3b| = √[(64)(1) + (24)(1/2) + (24)(1/2) + (9)(1)]
= √[64 + 12 + 12 + 9]
= √[97]
Therefore, |8a + 3b| = √[97] or approximately 9.849.
Hence, the magnitudes of the given expressions are |3a - 5b| ≈ 4.357 and |8a + 3b| ≈ 9.849.