Download App

Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors5 Marks
Question
The two vectors $\hat{\text{j}}+\hat{\text{k}}$ and $3\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}$ represents the sides $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$ respectively of a triangle ABC. Find the length of the median through A.
✓

Answer

Disclaimer: The question has been solved by taking the vector $\overrightarrow{\text{AB}}$ as $\hat{\text{j}}+\hat{\text{k}}$.In $\triangle\text{ABC},\ \overrightarrow{\text{AB}}=\hat{\text{j}}+\hat{\text{k}}$ and $\overrightarrow{\text{AC}}=3\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}$
Let the position vector of A be (0, 0, 0). Then, the position vectors of B and C are (0, 1, 1) and (3, -1, 4), respectively.

Suppose D be the mid-point of the line segment joining the points B(0, 1, 1) and C(3, -1, 4).
$\therefore$ position vector of D $=\frac{\big(\hat{\text{j}}+\hat{\text{k}}\big)+\big(3\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}\big)}{2}=\frac{3\hat{\text{i}}+5\hat{\text{k}}}{2}=\frac{3}2\hat{\text{i}}+\frac{5}2\hat{\text{k}}$
Now,
Length of the median, AD =
$\Big|\overrightarrow{\text{AD}}\Big|=\Big|\Big(\frac{3}2\hat{\text{i}}+\frac{5}2\hat{\text{k}}\Big)-\big(0\hat{\text{i}}+0\hat{\text{j}}+0\hat{\text{k}}\big)\Big|$
$=\Big|\frac{3}2\hat{\text{i}}+\frac{5}2\hat{\text{k}}\Big|$
$=\sqrt{\Big(\frac{3}{2}\Big)^2+0^2+\Big(\frac{5}2\Big)^2}$
$=\sqrt{\frac{34}{4}}$
$=\sqrt{\frac{17}2}\text{units}$

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

The two equal sides of an isosceles $\triangle$ with fixed base b are decreasing at the rate of $3 cm/s$. How fast is the area decreasing when the two equal sides are equal to the base?
→
There are two types of fertilisers 'A' and 'B'. 'A' consists of 12 % nitrogen and 5 % phosphoric acid whereas 'B' consists of 4 % nitrogen and 5 % phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs 10 per kg and 'B' cost 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost.
→
Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,$f(x) = -(x - 1)^3(x + 1), x < 1$
→
Prove that $2\tan^{-1}\bigg(\sqrt{\frac{\text{a}-\text{b}}{\text{a}+\text{b}}}\tan\frac{\theta}{2}\bigg)=\cos^{-1}\Big(\frac{\text{a}\cos\theta+b}{\text{a}+\text{b}\cos\theta}\Big)$
→
Find a particular solution of the differential equation $ \frac { d y } { d x } + y \cot x = 4 x \; cosec \; x$, x $\neq$ 0 given that y = 0, when $ x = \frac { \pi } { 2 }$.
→
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.
→
Prove that:
$\int\limits^\pi_0\text{xf}(\sin\text{x})\text{dx}=\frac{\pi}{2}\int\limits^\pi_0\text{f}(\sin\text{x})\text{dx}$
→
Solve the following differential equation: $\cos^{2} x \frac{dy}{dx} + y = \tan x$
→
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.
$f(x) = x(x - 1)$ on $[1, 2]$
→
If y(x) is a solution of $\Big(\frac{2+\sin\text{x}}{1+\text{y}}\Big)\frac{\text{dy}}{\text{dx}}=-\cos\text{x}$ and y(0) = 1, then find the value of $\text{y}\big(\frac{\pi}{2}\big).$
→