Find the value(s) of a making v =6ai−3j parallel to w =a^2i+9j .
a=
To be parallel
a^2/6a = 9/-3 = -3
a^2 = -18a
a^2 + 18a = 0
a(a+18) = 0
a = 0 or a = -18
check:
if a = 0 your vectors are <0,-3> and <0,9) which are parallel
if a = -18 your vectors are <-108,-3> and <324,9)
since <324,9> = -3<-108,-3> they are parallel
Sure, I can give you the value(s) of "a", but I must warn you, my answers might be clowning around a bit. Brace yourself!
Let's make v parallel to w. In other words, they should be like two synchronized circus performers, walking closely together. To achieve this, we need their directions to be the same.
First, let's extract the directions from each equation. For v, the direction is given by the coefficient of "i" and "j," so we have 6a for the "i" direction and -3 for the "j" direction.
For w, we have a^2 for the "i" direction and 9 for the "j" direction.
To make v parallel to w, we need the ratios of the directions to be the same. In other words, we need:
6a / -3 = a^2 / 9
Now, let's simplify and solve this equation:
2a / -1 = a^2 / 9
To avoid tricky negative signs, let's multiply both sides of the equation by -1:
2a = -a^2 / 9
Hmm, things are getting complicated! Let's simplify further:
2a = -a^2 / 9 * 2
Now, let's rewrite it again:
2a = -2a^2 / 9
Oh, division is not usually my specialty. Let's multiply both sides by 9 to get rid of the fraction:
18a = -2a^2
Now, my friend, it's time for solving the quadratic equation. Since math can be as serious as a clown with a frown, I will leave the solving to you. Good luck finding the value(s) of "a" that make v parallel to w!
To find the value(s) of a that make v parallel to w, we need to determine when the direction vectors for v and w are proportional.
The direction vector for v is given as 6ai − 3j, and the direction vector for w is given as a^2i + 9j.
Two vectors are proportional if their corresponding components are proportional. So, we can set up the following equation:
(6a) / (a^2) = (-3) / (9)
Simplifying this equation, we get:
2 / a = -1 / 3
Now, we can cross-multiply to solve for a:
2 * 3 = -1 * a
6 = -a
Divide both sides of the equation by -1 to solve for a:
a = -6
Therefore, the value of a that makes v parallel to w is -6.
To find the value(s) of "a" that make vector v parallel to vector w, we need to check if the two vectors are scalar multiples of each other.
Given:
v = 6ai - 3j
w = a^2i + 9j
Two vectors are scalar multiples of each other if and only if their corresponding components are proportional. In other words, the ratio of the corresponding components should be constant.
So, we can write the ratio of the i-components of the two vectors as:
(6a) / (a^2) = constant
To find the constant, we can simplify the ratio by canceling out the common factors:
6 / a = constant
If we let "k" be the constant, then we can rewrite the equation as:
6 / a = k
Now, we can solve for "a" by isolating it:
a = 6 / k
Therefore, the value(s) of "a" making vector v parallel to vector w are given by a = 6 / k, where "k" can be any non-zero constant.