RATIO FORMULA AND GEOMETRY - PUNJAB-CURRICULUM-AND-TEXTBOOK-BOARD-PCTB / 10TH-CLASS / MATHEMATICS / MATHEMATICS-10-10TH-CLASS-NBF-2024 / VECTORS-IN-PLANE / RATIO-FORMULA-AND-GEOMETRY

Ratio Formula for Finding Position Vector of a Point on a Vector

Let AB be any line segment and P be the point which divides this line segment in the given ratio m:n internally. The position vectors of the given points A and B are a and b respectively. Let r be the position vector of point P. Given that:

|\overrightarrow{AP}| : |\overrightarrow{PB}| = m:n

n\overrightarrow{AP} = m\overrightarrow{PB}

From figure, OA + AP = OP

\vec{a} + \vec{A P} = \vec{r} \Rightarrow \vec{A P} = \vec{r} - \vec{a}

Similarly, OP + PB = OB

\vec{r} + \vec{P B} = \vec{b} \Rightarrow \vec{P B} = \vec{b} - \vec{r}

Substituting the values, we get:

\vec{r} = \frac{m \vec{b} + n \vec{a}}{m + n}

Particular Case: If m = n, then P will be the midpoint of AB and position vector of P is:

\vec{r} = \frac{\vec{a} + \vec{b}}{2}

Example:

The position vectors of points A and B are 2i - j and 3i + 2j respectively. Find the position vector of point P dividing the line segment joining A and B in the ratio 3:4 internally.

Solution: Let a = 2i - j, b = 3i + 2j, and m:n = 3:4.

\vec{r} = \frac{m \vec{b} + n \vec{a}}{m + n} = \frac{3 (3 \vec{i} + 2 \vec{j}) + 4 (2 \vec{i} - \vec{j})}{7} = \frac{17}{7} \vec{i} + \frac{2}{7} \vec{j}

Application to Geometry

Example:

What type of a quadrilateral ABCD is if 2AB = DC?

Ratio Formula and Application to Geometry

Ratio Formula for Finding Position Vector of a Point on a Vector

Example:

Application to Geometry

Example: