Sectional Formula MCQ Quiz - Objective Question with Answer for Sectional Formula - Download Free PDF

Concept:

Section formula:

Let the position vector of A and B are $\overrightarrow{\alpha }$ and $\overrightarrow{\beta }$ respectively, and let P divide the line joining A and B in the ratio m : n internally.

Therefore, the position vector of P is given by $\frac{m\overrightarrow{\alpha }+n\overrightarrow{\beta }}{m+n}$.

Calculation:

Let position vector of points A, B, C be $\overrightarrow{a},\overrightarrow{b}$ & $\overrightarrow{c}$ respectively.

Also, let position vector of Q, P, and R be $\overrightarrow{0},\overrightarrow{p}$ and $\overrightarrow{r}$ respectively.

Using section formula

$\overrightarrow{a}=\frac{\overrightarrow{r}}{3},\overrightarrow{b}=\frac{\overrightarrow{p}+2\overrightarrow{r}}{3},\overrightarrow{c}=\frac{2\overrightarrow{p}}{3}$

Area of ΔPQR = $\left|\frac{1}{2}\right|\overrightarrow{\mathrm{QP}}\times \overrightarrow{\mathrm{QR}}||=\frac{1}{2}|\overrightarrow{r}\times \overrightarrow{p}|$

Now, Area of ΔABC = $\frac{1}{2}|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|$

= $\frac{1}{2}|(\frac{\overrightarrow{r}}{3}\times \left(\frac{\overrightarrow{p}+2\overrightarrow{r}}{3}\right))+\left(\frac{\overrightarrow{p}+2\overrightarrow{r}}{3}\right)\times \left(\frac{2\overrightarrow{p}}{3}\right)+(\frac{2\overrightarrow{p}}{3}\times \frac{\overrightarrow{r}}{3})|$

= $\frac{1}{2}|\frac{\overrightarrow{r}\times \overrightarrow{p}}{9}+4\left(\frac{\overrightarrow{r}\times \overrightarrow{p}}{9}\right)+\frac{2}{9}(\overrightarrow{p}\times \overrightarrow{r})|,\text{ as }\overrightarrow{r}\times \overrightarrow{r}=0$

= $\frac{1}{18}|\overrightarrow{r}\times \overrightarrow{p}+4(\overrightarrow{r}\times \overrightarrow{p})-2(\overrightarrow{r}\times \overrightarrow{p})|\text{ as }\overrightarrow{p}\times \overrightarrow{r}=-(\overrightarrow{r}\times \overrightarrow{p})$

= $\frac{|\overrightarrow{r}\times \overrightarrow{p}|}{6}$

So, $\frac{\text{ area of }\mathrm{△}\mathrm{PQR}}{\text{ area of }\mathrm{△}\mathrm{ABC}}$ = 3

∴ The value of $\frac{\text{ area of }\mathrm{△}\mathrm{PQR}}{\text{ area of }\mathrm{△}\mathrm{ABC}}$ is equal to 3.

The correct answer is Option 2.

Concept:

The position vector of the point that divides the line joining position vectors

$\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ in the ratio m:n is given by $\frac{m\overrightarrow{q}+n\overrightarrow{p}}{m+n}$.

Calculation:

Given position vectors are  $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$  and  $2\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$ .

∴ The position vector of the point which divides the line joining the above points in the ratio 1 : 2 is given by

 $=\frac{2(\overrightarrow{a}+\overrightarrow{b})+(2\overrightarrow{a}-\overrightarrow{b})}{2+1}$

=  $\frac{4\overrightarrow{a}+\overrightarrow{b}}{3}$

The position vector of the point which divides the join of points in the ratio 1 ∶ 2 is $\frac{4\overrightarrow{a}+\overrightarrow{b}}{3}$.

Concept:

The position vector of the point that divides the line joining position vectors

$\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ in the ratio m:n is given by $\frac{m\overrightarrow{q}+n\overrightarrow{p}}{m+n}$.

Calculation:

Given position vectors are  $2\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}$  and  $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ .

∴ The position vector of the point which divides the line joining the above points in the ratio 3: 1 is,

= $\frac{(2\overrightarrow{a}-3\overrightarrow{b})+3(\overrightarrow{a}+\overrightarrow{b})}{1+3}$

=  $\frac{5\overrightarrow{a}}{4}$

The position vector of the point which divides the join of points $2\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ in the ratio 3 ∶ 1 is $\frac{5\overrightarrow{a}}{4}$.

The correct answer is option 4.

Concept:

The position vector of the point that divides the line joining position vectors

$\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ in the ratio m:n is given by $\frac{m\overrightarrow{q}+n\overrightarrow{p}}{m+n}$.

Calculation:

Given position vectors are  $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$  and  $2\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$ .

∴ The position vector of the point which divides the line joining the above points in the ratio 1 : 2 is given by

 $=\frac{2(\overrightarrow{a}+\overrightarrow{b})+(2\overrightarrow{a}-\overrightarrow{b})}{2+1}$

=  $\frac{4\overrightarrow{a}+\overrightarrow{b}}{3}$

The position vector of the point which divides the join of points in the ratio 1 ∶ 2 is $\frac{4\overrightarrow{a}+\overrightarrow{b}}{3}$.

The correct answer is option 4.

Concept:

Point dividing the line joining points (x₁, y₁, z₁) and (x₂, y₂, z₂) in the ratio m : n is :

$(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n})$

Calculation:

Given points P and Q are $-2\bar{\mathrm{i}}-3\bar{\mathrm{j}}+\bar{\mathrm{k}}$ and   $3\bar{\mathrm{i}}+3\bar{\mathrm{j}}+2\bar{\mathrm{k}}$ respectively.

∴ Points P and Q in cartesian form are (- 2, - 3, 1) and (3, 3, 2)

Position vector of dividing point is $\frac{-9}{2}\bar{\mathrm{i}}-6\bar{\mathrm{j}}+\frac{1}{2}\bar{\mathrm{k}}$

∴ Point in cartesian form is ($-\frac{9}{2}$, - 6, $\frac{1}{2}$)

Let the ratio be k : 1

⇒ $-\frac{9}{2}$ = $\frac{3k-2}{k+1}$

⇒ -9k - 9 = 6k - 4

⇒ - 5 = 15k

⇒ k = $-\frac{1}{3}$

Hence the ratio is  - 1 : 3.

The correct answer is option (4).

Concept:

The position vector of the point that divides the line joining position vectors

$\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ in the ratio m:n is given by $\frac{m\overrightarrow{q}+n\overrightarrow{p}}{m+n}$.

Calculation:

Given position vectors are  $2\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}$  and  $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ .

∴ The position vector of the point which divides the line joining the above points in the ratio 3: 1 is,

= $\frac{(2\overrightarrow{a}-3\overrightarrow{b})+3(\overrightarrow{a}+\overrightarrow{b})}{1+3}$

=  $\frac{5\overrightarrow{a}}{4}$

The position vector of the point which divides the join of points $2\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ in the ratio 3 ∶ 1 is $\frac{5\overrightarrow{a}}{4}$.

The correct answer is option 4.

Concept:

1. Let the given two position vector p and q respectively and r be the vector dividing the segment PQ internally in the ratio m: n

1. Internal Section Formula: When the segment PQ is divided internally in the ration m: n, we use this formula. ⇔ $\mathrm{r}=\left(\frac{\mathrm{m}\mathrm{q}+\mathrm{n}\mathrm{p}}{\mathrm{m}+\mathrm{n}}\right)$
2. External Section Formula: When the segment PQ is divided externally in the ratio m: n, we use this formula. ⇔ $\mathrm{s}=\left(\frac{\mathrm{m}\mathrm{q}-\mathrm{n}\mathrm{p}}{\mathrm{m}-\mathrm{n}}\right)$

2. $\overrightarrow{\mathrm{a}}\mathrm{a}\mathrm{n}\mathrm{d}\overrightarrow{\mathrm{b}}$ are two vectors perpendicular to each other ⇔ $\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}=0$

Calculation:

Given that,

Vector R divide segment PQ internally in ration 2 : 3 so,

Form the diagram,

$\Rightarrow \mathrm{R}=\left(\frac{2\mathrm{q}+3\mathrm{p}}{2+3}\right)$

$\Rightarrow \mathrm{R}=\frac{\left(2\mathrm{q}+3\mathrm{p}\right)}{5}$       …(1)

Given Vector S divide segment PQ internally in ration 2 : 3 so,

$\Rightarrow \mathrm{S}=\left(\frac{2\mathrm{q}-3\mathrm{p}}{2-3}\right)$

$\Rightarrow \mathrm{S}=\frac{3\mathrm{p}-2\mathrm{q}}{1}$       …(2)

Note: given vectors are position vectors with respect to O so vector OR = R and OS = s

Given OR and OS perpendicular to each other so,

⇒ R.S = 0

From equation 1 and 2

$\Rightarrow \frac{\left(2\mathrm{q}+3\mathrm{p}\right)}{5}.\frac{3\mathrm{p}-2\mathrm{q}}{1}=0$

⇒ 6q.p - 4q² + 9p² - 6p.q = 0

⇒ 4q² = 9p²

Concept:

The position vector of the point that divides the line joining position vectors

$\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ in the ratio m:n is given by $\frac{m\overrightarrow{q}+n\overrightarrow{p}}{m+n}$.

Calculation:

Given position vectors are  $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$  and  $2\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$ .

∴ The position vector of the point which divides the line joining the above points in the ratio 1 : 2 is given by

 $=\frac{2(\overrightarrow{a}+\overrightarrow{b})+(2\overrightarrow{a}-\overrightarrow{b})}{2+1}$

=  $\frac{4\overrightarrow{a}+\overrightarrow{b}}{3}$

The position vector of the point which divides the join of points in the ratio 1 ∶ 2 is $\frac{4\overrightarrow{a}+\overrightarrow{b}}{3}$.

The correct answer is option 4.

Concept:

For internal division: P divides the AB internally in ratio m : n

P(x) = $\frac{\mathrm{m}{\mathrm{x}}_2+\mathrm{n}{\mathrm{x}}_1}{\mathrm{m}+\mathrm{n}}$

Calculation:

Let us consider x be the position vector of B, then P divides AB is the ratio 2 : 3

a = $\frac{2\mathrm{x}+3(\mathrm{a}+2\mathrm{b})}{2+3}$

5a = 2x + 3a + 6b

2x = 2a - 6b

x = a - 3b

 The position vector of B is a - 3b.

Concept:

1. Let the given two position vectors p and q respectively and r be the vector dividing the segment PQ internally in the ratio m: n

1. Internal Section Formula: When the segment PQ is divided internally in the ratio m: n, we use this formula. ⇔ $\mathrm{r}=\left(\frac{\mathrm{m}\mathrm{q}+\mathrm{n}\mathrm{p}}{\mathrm{m}+\mathrm{n}}\right)$
2. External Section Formula: When the segment PQ is divided externally in the ratio m: n, we use this formula. ⇔ $\mathrm{s}=\left(\frac{\mathrm{m}\mathrm{q}-\mathrm{n}\mathrm{p}}{\mathrm{m}-\mathrm{n}}\right)$

Calculation:

Let the required ratio is k :1 then the point suing section formula of the line joining (-2, 4, 7) and (3, -5, 8),

$(\frac{3k-2}{k+1},\frac{-5k+4}{k+1},\frac{8k+7}{k+1})$

The above point will lie on the plane so,

$\Rightarrow 1(\frac{3k-2}{k+1})-2(\frac{-5k+4}{k+1})+3(\frac{8k+7}{k+1})=17$

⇒ k = 3/10

Concept:

The position vector of the point that divides the line joining position vectors

$\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ in the ratio m:n is given by $\frac{m\overrightarrow{q}+n\overrightarrow{p}}{m+n}$.

Calculation:

Given position vectors are  $2\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}$  and  $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ .

∴ The position vector of the point which divides the line joining the above points in the ratio 3: 1 is,

= $\frac{(2\overrightarrow{a}-3\overrightarrow{b})+3(\overrightarrow{a}+\overrightarrow{b})}{1+3}$

=  $\frac{5\overrightarrow{a}}{4}$

The position vector of the point which divides the join of points $2\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ in the ratio 3 ∶ 1 is $\frac{5\overrightarrow{a}}{4}$.

The correct answer is option 4.

Concept:

1. Let the given two position vector p and q respectively and r be the vector dividing the segment PQ internally in the ratio m: n

1. Internal Section Formula: When the segment PQ is divided internally in the ration m: n, we use this formula. ⇔ $\mathrm{r}=\left(\frac{\mathrm{m}\mathrm{q}+\mathrm{n}\mathrm{p}}{\mathrm{m}+\mathrm{n}}\right)$
2. External Section Formula: When the segment PQ is divided externally in the ratio m: n, we use this formula. ⇔ $\mathrm{s}=\left(\frac{\mathrm{m}\mathrm{q}-\mathrm{n}\mathrm{p}}{\mathrm{m}-\mathrm{n}}\right)$

2. $\overrightarrow{\mathrm{a}}\mathrm{a}\mathrm{n}\mathrm{d}\overrightarrow{\mathrm{b}}$ are two vectors perpendicular to each other ⇔ $\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}=0$

Calculation:

Given that,

Vector R divide segment PQ internally in ration 2 : 3 so,

Form the diagram,

$\Rightarrow \mathrm{R}=\left(\frac{2\mathrm{q}+3\mathrm{p}}{2+3}\right)$

$\Rightarrow \mathrm{R}=\frac{\left(2\mathrm{q}+3\mathrm{p}\right)}{5}$       …(1)

Given Vector S divide segment PQ internally in ration 2 : 3 so,

$\Rightarrow \mathrm{S}=\left(\frac{2\mathrm{q}-3\mathrm{p}}{2-3}\right)$

$\Rightarrow \mathrm{S}=\frac{3\mathrm{p}-2\mathrm{q}}{1}$       …(2)

Note: given vectors are position vectors with respect to O so vector OR = R and OS = s

Given OR and OS perpendicular to each other so,

⇒ R.S = 0

From equation 1 and 2

$\Rightarrow \frac{\left(2\mathrm{q}+3\mathrm{p}\right)}{5}.\frac{3\mathrm{p}-2\mathrm{q}}{1}=0$

⇒ 6q.p - 4q² + 9p² - 6p.q = 0

⇒ 4q² = 9p²

Concept:

The position vector of the point that divides the line joining position vectors

$\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ in the ratio m:n is given by $\frac{m\overrightarrow{q}+n\overrightarrow{p}}{m+n}$.

Calculation:

Given position vectors are  $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$  and  $2\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$ .

∴ The position vector of the point which divides the line joining the above points in the ratio 1 : 2 is given by

 $=\frac{2(\overrightarrow{a}+\overrightarrow{b})+(2\overrightarrow{a}-\overrightarrow{b})}{2+1}$

=  $\frac{4\overrightarrow{a}+\overrightarrow{b}}{3}$

The position vector of the point which divides the join of points in the ratio 1 ∶ 2 is $\frac{4\overrightarrow{a}+\overrightarrow{b}}{3}$.

The correct answer is option 4.

Concept:

Point dividing the line joining points (x₁, y₁, z₁) and (x₂, y₂, z₂) in the ratio m : n is :

$(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n})$

Calculation:

Given points P and Q are $-2\bar{\mathrm{i}}-3\bar{\mathrm{j}}+\bar{\mathrm{k}}$ and   $3\bar{\mathrm{i}}+3\bar{\mathrm{j}}+2\bar{\mathrm{k}}$ respectively.

∴ Points P and Q in cartesian form are (- 2, - 3, 1) and (3, 3, 2)

Position vector of dividing point is $\frac{-9}{2}\bar{\mathrm{i}}-6\bar{\mathrm{j}}+\frac{1}{2}\bar{\mathrm{k}}$

∴ Point in cartesian form is ($-\frac{9}{2}$, - 6, $\frac{1}{2}$)

Let the ratio be k : 1

⇒ $-\frac{9}{2}$ = $\frac{3k-2}{k+1}$

⇒ -9k - 9 = 6k - 4

⇒ - 5 = 15k

⇒ k = $-\frac{1}{3}$

Hence the ratio is  - 1 : 3.

The correct answer is option (4).

Concept:

In the yz-plane, the x coordinate is always 0.

To find the ratio, we assume it λ : 1.

On solving with the given conditions, if λ is positive then ratio is internal otherwise external.

Formula:

If the join of two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ is divided in the ratio m : n by a point P (x , y , z), then the coordinates of the point P will be -

$x=\frac{m{x}_2+n{x}_1}{m+n}$ , $y=\frac{m{y}_2+n{y}_1}{m+n}$  , $z=\frac{m{z}_2+n{z}_1}{m+n}$

Calculation:

Given:

Points A(4, 8, 10) and B(6, 10, -8)

By evaluating x-coordinate,

⇒ $0=\frac{4+6\lambda }{\lambda +1}$

⇒ 4 + 6λ = 0 

⇒ λ = -2/3   (negative sign means externally)