Download App

Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors5 Marks
Question
Five forces $\overrightarrow{\text{AB}},\ \overrightarrow{\text{AC}},\ \overrightarrow{\text{AD}},\ \overrightarrow{\text{AE}}\text{ and }\overrightarrow{\text{AF}}$ act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is:$6\ \overrightarrow{\text{AO}}$, where o is the center of hexagon.
✓

Answer


$\overrightarrow{\text{AB}}+\overrightarrow{\text{AC}}+\overrightarrow{\text{AD}}+\overrightarrow{\text{AE}}+\overrightarrow{\text{AF}}$
Consider $\triangle\text{ADE}$,
$\overrightarrow{\text{AD}}+\overrightarrow{\text{DE}}+\overrightarrow{\text{EA}}=0$
$\overrightarrow{\text{AD}}+\overrightarrow{\text{DE}}=\overrightarrow{\text{AE}}$
$2\ \overrightarrow{\text{AO}}+\Big(-\overrightarrow{\text{AB}}\Big)=\overrightarrow{\text{AE}}$ $\Big[\overrightarrow{\text{AD}}=2\ \overrightarrow{\text{AO}}\text{ and }\overrightarrow{\text{ED}}\ \Big|\Big|\ \overrightarrow{\text{AB}}+\overrightarrow{\text{DE}}=-\overrightarrow{\text{AB}}\Big]$
$\therefore\ \overrightarrow{\text{AE}}+\overrightarrow{\text{AB}}=2\ \overrightarrow{\text{AO}}\ \dots(1)$
Now, consider $\triangle\text{ADC}$,
$\overrightarrow{\text{AC}}+\overrightarrow{\text{CD}}+\overrightarrow{\text{DA}}=0$
$\overrightarrow{\text{AC}}+\overrightarrow{\text{CD}}=\overrightarrow{\text{AD}}$ $\Big[\because\ \overrightarrow{\text{CD}}=\overrightarrow{\text{AF}}\Big]$
$\overrightarrow{\text{AC}}+\overrightarrow{\text{AF}}=2\ \overrightarrow{\text{AO}}\ \dots(2)$
Using (1) and (2),
$\overrightarrow{\text{AB}}+\overrightarrow{\text{AE}}+\overrightarrow{\text{AC}}+\overrightarrow{\text{AF}}+\overrightarrow{\text{AD}}$
$2\ \overrightarrow{\text{AO}}+2\ \overrightarrow{\text{AO}}+2\ \overrightarrow{\text{AO}}$
$=6\ \overrightarrow{\text{AO}}$

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 "5 Marks Questions" from Algebra of Vectors→Algebra of Vectors — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int(\text{x}=1)\sqrt{\text{x}^2-\text{x}+1}\text{dx}$
→
Solve the following differential equation
$\sin^4\text{x}\frac{\text{dy}}{\text{dx}}=\cos\text{x}$
→
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
f(x) = (x - 1)(x - 2)(x - 3) on [0, 4]
→
If $\text{A} = \begin{pmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 &1 & -2 \end{pmatrix}$ find $A^{–1}$. Hence using $A^{–1}$ solve the system of equations $2x – 3y + 5z = 11, 3x + 2y – 4z = – 5, x + y – 2z = – 3$.
→
Show that the semi $-$ vertical angle of a cone of maximum volume and given slant height is $\tan ^{-1} \sqrt{2}$ or $\cos ^{-1} \frac{1}{\sqrt{3}}$.
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 2x^2 - 24x + 7$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{6}}_{0}\cos^{-3}2\theta\sin2\theta\text{ d}\theta$
→
Find the equation of the plane through the line of intersection of the planes $x + 2y + 3z + 4 = 0$ and $x - y + z + 3 = 0$ and passing through the origin.
→
Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\sqrt{\sin\phi}\cos^5\phi\text{ d}\phi$
→
A gardener has supply of fertilizer of type I which consists of 10% nitrogen and 6% phosphoric acid and type II fertilizer which consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, he finds that he needs at least 14kg of nitrogen and 14kg of phosphoric acid for his crop. If the type I fertilizer costs 60 paise per kg and type II fertilizer costs 40 paise per kg, determine how many kilograms of each fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?
→