Two vectors are given by
a = 3.7i + 6.9j, and
b = 1.9i + 4.7j.
Find
(b) a·b =
(c) (a + b)·b =
(d) the component of a in the direction of b =
.
To find the dot product of two vectors, we multiply their corresponding components and then sum them up.
b) To find a·b, we can multiply the corresponding components of vectors a and b and then sum them up:
a·b = (3.7 * 1.9) + (6.9 * 4.7)
= 7.03 + 32.43
= 39.46
So, a·b = 39.46.
c) To find (a + b)·b, we first need to find the sum of vectors a and b. We can add their corresponding components:
(a + b) = (3.7 + 1.9)i + (6.9 + 4.7)j
= 5.6i + 11.6j
Now, we can find the dot product of (a + b) and b:
(a + b)·b = (5.6 * 1.9) + (11.6 * 4.7)
= 10.64 + 54.52
= 65.16
So, (a + b)·b = 65.16.
d) To find the component of vector a in the direction of vector b, we can use the formula:
component of a in the direction of b = (a·b) / |b|
First, we need to find the magnitude (or length) of vector b. The magnitude of a vector is given by the square root of the sum of the squares of its components:
|b| = sqrt((1.9)^2 + (4.7)^2)
= sqrt(3.61 + 22.09)
= sqrt(25.7)
= 5.07
Now, we can calculate the component of a in the direction of b:
component of a in the direction of b = (a·b) / |b|
= 39.46 / 5.07
= 7.79
So, the component of a in the direction of b is 7.79.