Download App

VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA3 Marks
Question
ABCDE is a pentagon, prove that,$\overrightarrow{\text{AB}}+\overrightarrow{\text{AE}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{DC}}+\overrightarrow{\text{ED}}+\overrightarrow{\text{AC}}=3\ \overrightarrow{\text{AC}}$
✓

Answer

It is given that ABCDE is a pentagon, So
$\overrightarrow{\text{AB}}+\overrightarrow{\text{AE}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{DC}}+\overrightarrow{\text{ED}}+\overrightarrow{\text{AC}}$
$=\Big(\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}\Big)+\overrightarrow{\text{AE}}+\overrightarrow{\text{DC}}+\overrightarrow{\text{ED}}+\overrightarrow{\text{AC}}$
$=\overrightarrow{\text{AC}}+\overrightarrow{\text{DC}}+\Big(\overrightarrow{\text{AE}}+\overrightarrow{\text{ED}}\Big)+\overrightarrow{\text{AC}}$ $\Big[$Using triangle law in $\triangle\text{ABC},\ \overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}=\overrightarrow{\text{AC}}\Big]$
$=\overrightarrow{\text{AC}}+\overrightarrow{\text{DC}}+\Big(\overrightarrow{\text{AD}}\Big)+\overrightarrow{\text{AC}}$
$=\overrightarrow{\text{AC}}+\overrightarrow{\text{DC}}-\overrightarrow{\text{DA}}+\overrightarrow{\text{AC}}$
$=\overrightarrow{\text{AC}}+\overrightarrow{\text{DC}}+\overrightarrow{\text{AD}}+\overrightarrow{\text{AC}}$
$=\overrightarrow{\text{AC}}+\overrightarrow{\text{AD}}+\overrightarrow{\text{DC}}+\overrightarrow{\text{AC}}$
$=\overrightarrow{\text{AC}}+\overrightarrow{\text{AC}}+\overrightarrow{\text{AC}}$
$=3\ \overrightarrow{\text{AC}}$
So,
$\overrightarrow{\text{AB}}+\overrightarrow{\text{AE}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{DC}}+\overrightarrow{\text{ED}}+\overrightarrow{\text{AC}}=3\ \overrightarrow{\text{AC}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from VECTOR ALGEBRA→VECTOR ALGEBRA — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

On the set Z of integers a binary operation * is defined by a * b = ab + 1 for all a, b ∈ Z. Prove that * is not associative on Z.
→
Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\sin\text{x }\text{dx}$
→
Form the differential equation having $\text{y} = (\sin^{-1}\text{x})^{2} + \text{A}\cos^{-1} + \text{ B},$ where A and B are arbitrary constants, as its general solution.
→
show that $\text{f}\text{(x)}=\begin{cases}\frac{\text{x}-|\text{x}|}{2}, & \text{when} \text{ x}\neq 0\\2, & \text{when}\text{ x} = 0\end{cases}$ is discontinuous at x = 0.
→
Find the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are
$\vec r = \hat i + \hat j + \lambda (2\hat i - \hat j + \hat k)$ ...(1)
and $\vec r = 2\hat i + \hat j - \hat k + \mu (3\hat i - 5\hat j + 2\hat k)$ ...(2)
→
Evaluate the following integrals:
$\int\cot^{\text{n}}\text{cosec}^2\text{x}\text{ dx},\text{ n}\neq-1$
→
Find the volume of the parallelopiped whose coterminous edges are represented by the vectors:
$\vec{\text{a}}=11\hat{\text{i}},\vec{\text{b}}=2\hat{\text{j}},\vec{\text{c}}=13\hat{\text{k}}$
→
Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes $\vec{\text{r}}.\Big(\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}\Big)=5\ \text{and}\ \vec{\text{r}}.\Big(3\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\Big)=6.$
→
The probability of student A passing an examination is $\frac{2}{9}$ and of student B passing is $\frac{5}{9}$. Assuming the two events: 'A passes', 'B passes' as independent, find the probability of:
Only one of them passing the examination.
→
Solve the following differential equation:
$(y^2 - x^2) dy = 3xy\ dx$.
→