Find the value(s) of a making v =5ai −2j parallel to w =a2i +2j .
you need a^2/5a = 2/-2
To find the value(s) of a that make vector v parallel to vector w, we can compare the direction ratios of the two vectors.
The direction ratio of a vector is the coefficients of its components.
For vector v = 5ai - 2j, the direction ratio is (5a, -2).
And for vector w = a^2i + 2j, the direction ratio is (a^2, 2).
If v is parallel to w, then their direction ratios should be proportional. In other words, we need to find the value(s) of a that satisfy the proportionality relation:
(5a, -2) = k(a^2, 2)
where k is a constant.
Setting the corresponding components equal to each other, we get two equations:
5a = ka^2 (equation 1)
-2 = 2k (equation 2)
From equation 2, we can solve for k:
2k = -2
k = -1
Substituting k = -1 into equation 1:
5a = (-1)a^2
Now, there are two cases to consider:
Case 1: a = 0
If a = 0, then vector w becomes w = 0i + 2j = 2j.
In this case, v will be parallel to w for any value of a.
Case 2: a ≠ 0
If a ≠ 0, then we can divide both sides of the equation by a:
5 = -a
Simplifying further, we get:
a = -5
Therefore, the value of a that makes v = 5ai - 2j parallel to w = a^2i + 2j is a = -5.
To find the value(s) of "a" that make vector v parallel to vector w, we need to check if the direction vectors of the two vectors are proportional.
Let's break down the given vectors into their components:
v = 5ai - 2j (1)
w = a^2i + 2j (2)
From (1), we can see that the x-component of vector v is 5a, and the y-component is -2.
From (2), we can see that the x-component of vector w is a^2, and the y-component is 2.
For vector v and vector w to be parallel, the ratio of their x-components must be equal to the ratio of their y-components.
So we set up the following proportion:
(5a) / (a^2) = (-2) / (2)
Simplifying this proportion, we get:
5a / a^2 = -2 / 2
Dividing both sides by "a" (assuming a ≠ 0) gives us:
5 / a = -1
Now, we can solve for "a" by cross-multiplying:
5 = -a
Dividing both sides by -1 gives us:
a = -5
So, the value of "a" that makes vector v parallel to vector w is a = -5.