Download App

Scalar Or Dot Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product3 Marks
Question
If $|\vec{\text{a}}|=\text{a}$ and $\big|\vec{\text{b}}\big|=\text{b},$ prove that $\Big(\frac{\vec{\text{a}}}{\text{a}^2}-\frac{\vec{\text{b}}}{\text{b}^2}\Big)^2=\Big(\frac{\vec{\text{a}}-\vec{\text{b}}}{\text{ab}}\Big)^2.$
✓

Answer

$\Big(\frac{\vec{\text{a}}}{\text{a}^2}-\frac{\vec{\text{b}}}{\text{b}^2}\Big)^2$
$=\Big|\frac{\vec{\text{a}}}{\text{a}^2}\Big|^2+\Big|\frac{\vec{\text{b}}}{\text{b}^2}\Big|^2=\frac{2\vec{\text{a}}.\vec{\text{b}}}{\text{a}^2\text{b}^2}$
$=\frac{|\vec{\text{}a}|^2}{\text{a}^4}+\frac{\big|\vec{\text{b}}\big|^2}{\text{b}^4}-\frac{2\vec{\text{a}}.\vec{\text{b}}}{\text{a}^2\text{b}^2}$
$=\frac{\text{a}^2}{\text{a}^4}+\frac{\text{b}^2}{\text{b}^4}-\frac{2\vec{\text{a}}.\vec{\text{b}}}{\text{a}^2\text{b}^2}$ (From the given information)
$=\frac{1}{\text{a}^2}+\frac{1}{\text{b}^2}-\frac{2\vec{\text{a}}.\vec{\text{b}}}{\text{a}^2\text{b}^2}$
$=\frac{\text{b}^2+\text{a}^2-2\vec{\text{a}}.\vec{\text{b}}}{\text{a}^2\text{b}^2}$
$=\frac{\text{a}^2+\text{b}^2-2\vec{\text{a}}.\vec{\text{b}}}{\text{a}^2\text{b}^2}$
$=\frac{|\vec{\text{a}}|^2+|\vec{\text{a}}|^2-2\vec{\text{a}}.\vec{\text{b}}}{\text{a}^2\text{b}^2}$ (From the given information)
$=\frac{\big(\vec{\text{a}}-\vec{\text{b}}\big)^2}{\text{a}^2\text{b}^2}$
$=\Big(\frac{\vec{\text{a}}-\vec{\text{b}}}{\text{ab}}\Big)^2$

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" from Scalar Or Dot Product→Scalar Or Dot Product — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If the mean and variance of a binomial distribution are respectively 9 and 6, find the distribution.
→
Show that $\text{f(x)}=2\text{x}+\cot^{-1}\text{x}+\log\Big(\sqrt{1+\text{x}^1}-\text{x}\Big)$ is increasing in R.
→
Find the angle between the plane:
x - y + z = 5 and x + 2y + z = 9
→
If $\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}},$ and $\vec{\text{b}}=2\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}},$ then find $\vec{\text{a}}\times\vec{\text{b}}.$ verify that $\vec{\text{a}}$ and $\vec{\text{a}}\times\vec{\text{b}}$ are perpendicular to each other.
→
Show that the function given by f(x) = $\frac{\log x}{x}$ has maximum at x = e.
→
Let the function f : R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto.
→
A fair coin and an unbiased die are tossed. Let A be the event head appears on the coin and B be the event 3 on the die. Check whether A and B are independent events or not.
→
Find the value of k for which $\text{f(x)}=\begin{cases}\frac{1-\cos4\text{x}}{8\text{x}^2},&\text{when x}\neq0\\\text{k},&\text{when x}=0\end{cases}$ is continous at x = 0.
→
A coin is tossed 5 times. If X is the number of heads observed, find the probability distribution of X.
→
Write a value of $\int\frac{1+\cot\text{x}}{\text{x}+\log\sin\text{x}}\text{ dx}$
→