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
To find the value of x that satisfies the given conditions, we can compare the components of the two vectors and set up equations based on the given conditions.
a) If the two vectors, a and b, are parallel, then their direction ratios will be proportional. In other words, the ratio of the corresponding components will be the same.
Comparing the x-components, we have:
2 = 1 * x
Solving the equation gives:
x = 2
So, if the vectors are parallel, the value of x is 2.
b) If the two vectors, a and b, are perpendicular, then their dot product will be zero. The dot product is calculated as the sum of the product of the corresponding components.
Comparing the dot product of the two vectors, we have:
a · b = 2 * 1 + x * (-4) = 0
Simplifying the equation gives:
2 - 4x = 0
Solving for x, we have:
4x = 2
x = 2/4
x = 1/2
So, if the vectors are perpendicular, the value of x is 1/2.