two vectors are defined as a=2i+xj and b=i-4j. find value of x if
a) the vectors are parallel
b) the vectors are perpendicular
dy/dx = x/2
dy/dx = -4
if parallel x/2 = -4 so x = -8
if perpendicular x/2 = - (1/-4) = 1/4
so x = 1/2
a) To determine if two vectors are parallel, we need to compare their direction ratios. In this case, the given vectors are a = 2i + xj and b = i - 4j.
For two vectors to be parallel, their direction ratios should be proportional.
Therefore, we can set up the following equations by comparing the components of vector a and vector b:
2 = 1 * k (Comparing the x-components)
x = -4 * k (Comparing the y-components)
Where k is the proportionality constant.
From the first equation, we have:
2 = k
Substituting k in the second equation, we get:
x = -4 * 2
x = -8
Therefore, the value of x is -8 if the vectors are parallel.
b) To determine if two vectors are perpendicular, we can use the dot product.
The dot product of two perpendicular vectors is zero.
Let's calculate the dot product of a = 2i + xj and b = i - 4j:
a · b = (2i + xj) · (i - 4j)
= 2 * 1 + x * (-4)
= 2 - 4x
Since the dot product of two perpendicular vectors is zero, we have:
2 - 4x = 0
Solving the equation for x, we get:
4x = 2
x = 1/2
Therefore, the value of x is 1/2 if the vectors are perpendicular.
To determine if two vectors are parallel or perpendicular, we can use the dot product. The dot product of two vectors is given by the formula:
A · B = |A| |B| cos(θ)
where A and B are the vectors, |A| and |B| are their magnitudes, and θ is the angle between them.
a) To find the value of x if the vectors are parallel, we need to calculate the dot product between the two vectors and set it equal to the product of their magnitudes:
a · b = |a| |b|
So, let's calculate the dot product:
a · b = (2i + xj) · (i - 4j)
= 2 * 1 + x * (-4)
= 2 - 4x
The magnitudes of the vectors are:
|a| = √(2^2 + x^2) = √(4 + x^2)
|b| = √(1^2 + (-4)^2) = √(1 + 16) = √17
Setting the dot product equal to the product of their magnitudes:
2 - 4x = √(4 + x^2) * √17
To solve this equation for x, we need to square both sides:
(2 - 4x)^2 = (4 + x^2) * 17
4 - 16x + 16x^2 = 68 + 17x^2
16x^2 - 17x^2 - 16x + 4 = 68
-x^2 - 16x - 64 = 0
Now we can solve this quadratic equation to find the value(s) of x.
b) To find the value of x if the vectors are perpendicular, we need to calculate the dot product between the two vectors and set it equal to zero:
a · b = 0
Using the same dot product as before:
a · b = (2i + xj) · (i - 4j)
= 2 * 1 + x * (-4)
= 2 - 4x
Setting the dot product equal to zero:
2 - 4x = 0
Solving this equation will give us the value of x when the vectors are perpendicular.