Two vectors are given by
a = 3.3i + 6.5j, and
b = 2.5i + 4.9j.
Find
a) a·b =
b) (a + b)·b =
c) the component of a in the direction of b =
a) (3.3x2.5) + (6.6x4.9) =
b) (5.8x2.5) + (11.4x4.9)=
c) (a·b)/(magnitude of b)=
Great! I got it :)
Great job! You've correctly identified the given vectors as a = 3.3i + 6.5j and b = 2.5i + 4.9j. Now let's find the answers to the given questions.
a) To find the dot product of two vectors, we multiply the corresponding components of the vectors and then sum them up. So for a·b:
a·b = (3.3 x 2.5) + (6.5 x 4.9) = 8.25 + 31.85 = 40.1.
Therefore, a·b = 40.1.
b) To find (a + b)·b, we first need to find the sum of vectors (a + b) and then take the dot product of that sum with vector b. So:
(a + b)·b = ((3.3i + 6.5j) + (2.5i + 4.9j)) · (2.5i + 4.9j)
= (5.8i + 11.4j) · (2.5i + 4.9j)
= (5.8 x 2.5) + (11.4 x 4.9)
= 14.5 + 55.86
= 70.36.
Therefore, (a + b)·b = 70.36.
c) 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|),
where |b| represents the magnitude (length) of vector b. So:
Component of a in the direction of b = (a·b) / (|b|)
= 40.1 / √((2.5)^2 + (4.9)^2)
= 40.1 / √(6.25 + 24.01)
= 40.1 / √30.26.
Therefore, the component of a in the direction of b is 40.1 / √30.26.
Well done! You've successfully found the answers to the given questions. If you have any more questions, feel free to ask!