Given that position vector a=4i+2j, position vector b=13i+5j are the vectors of A and B respectively. Given also that C lies on the line AB such that (Ab/AC)=3.
Find vector AB, vector AC and the position vector of C.
Given that vector BD=2i+4j, show that D lies on the line AB ad find the length of OD.
Can someone help me to solve it please?
To solve these problems, we will make use of vector algebra and properties.
1. Find vector AB, vector AC, and the position vector of C:
Vector AB can be found by subtracting vector A from vector B:
AB = vector B - vector A = (13i + 5j) - (4i + 2j) = 13i + 5j - 4i - 2j = 9i + 3j
To find vector AC, we can use the fact that (AB/AC) = 3. This means that the magnitude of AB is 3 times the magnitude of AC:
|AB| = 3|AC|
The magnitude of a vector AB is given by |AB| = sqrt(Ax^2 + Ay^2), where Ax and Ay are the components of vector AB in the x and y directions respectively.
Since we already know that |AB| = sqrt(9^2 + 3^2), we can solve for |AC| as follows:
sqrt(9^2 + 3^2) = 3|AC|
sqrt(90) = 3|AC|
|AC| = sqrt(90) / 3 = sqrt(10)
Now, let's calculate the position vector of C. We can use the following formula: vector C = vector A + (|AC| / |AB|) * vector AB
= (4i + 2j) + (sqrt(10) / sqrt(90)) * (9i + 3j)
= (4i + 2j) + (sqrt(10) / 3) * (9i + 3j)
= (4i + 2j) + (3sqrt(10) / 3) * (9i + 3j)
= (4i + 2j) + sqrt(10) * (3i + j)
= (4 + 3sqrt(10))i + (2 + sqrt(10))j
Therefore, the position vector of C is (4 + 3sqrt(10))i + (2 + sqrt(10))j.
2. To show that point D lies on the line AB and find the length of OD:
To show that D lies on the line AB, we need to prove that the vector AB and vector BD are parallel.
If two vectors are parallel, their direction ratios will be proportional.
Let's find the direction ratios of vector AB and vector BD:
Direction ratios of AB: (9, 3)
Direction ratios of BD: (2, 4)
To check if they are proportional, divide the corresponding components:
9/2 = 3/4
Since the ratio is the same, we can conclude that vector AB and vector BD are parallel. Therefore, point D lies on the line AB.
To find the length of OD, we can use the formula |OD| = sqrt(Dx^2 + Dy^2), where Dx and Dy are the components of vector OD in the x and y directions respectively.
We already know that OD = vector BD = 2i + 4j. So, we can substitute the values into the formula:
|OD| = sqrt((2)^2 + (4)^2) = sqrt(4 + 16) = sqrt(20) = 2sqrt(5)
Therefore, the length of OD is 2sqrt(5).