Vector Representation in a Triangle
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Q.2. In $\triangle PQR$, $\vec{PQ} = 2\vec{a}$, $\vec{QR} = 2\vec{b}$. The midpoint of $\vec{PR}$ is $M$. Find following vectors in terms of $\vec{a}, \vec{b}$. 1) $\vec{PR}$ 2) $\vec{PM}$ 3) $\vec{QM}$
This question includes visual content: A hand-drawn triangle labeled P, Q, R. The side PQ is marked with a vector arrow pointing from P to Q, labeled '2a'. The side QR is marked with a vector arrow pointing from Q to R, labeled '2b'. A line segment connects vertex Q to point M on the side PR. Point M is indicated as the midpoint of PR.
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Hi Tanishka, let's solve this vector geometry problem step-by-step.
Vector Relations in a Triangle
We are given that the vector from P to Q is two a, and the vector from Q to R is two b. M is the midpoint of side PR. We want to find vectors PR, PM, and QM in terms of a and b.
First, let's find the vector P R. By the triangle law of vector addition, the vector from P to R is equal to the sum of the vector from P to Q and the vector from Q to R.
Part 1: Finding Vector PR
Now, we substitute the given vector P Q which is two a, and vector Q R which is two b.
So we have our first result, vector P R is equal to two a plus two b.
Next, let's find vector P M. We are given that M is the midpoint of segment P R.
Part 2: Finding Vector PM
Since $M$ is the midpoint of $PR$:
Since vector P M is half of vector P R, we can substitute our previous result for P R into this equation.
Distributing the factor of one half, we get vector P M equals vector a plus vector b.
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