Download App

Scalar Or Dot Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product5 Marks
Question
If $\big|\vec{\text{a}}+\vec{\text{b}}\big|=60,\big|\vec{\text{a}}-\vec{\text{b}}\big|=40$ and $\big|\vec{\text{b}}\big|=46,$ find $|\vec{\text{a}}|$
✓

Answer

Here, $\big|\vec{\text{a}}+\vec{\text{b}}\big|=60$
Squaring both the sides,
$\big|\vec{\text{a}}+\vec{\text{b}}\big|^2=(60)^2$
$(\vec{\text{a}}+\vec{\text{b}})=(60)^2$
$(\vec{\text{a}})^2+(\vec{\text{b}})^2+2\vec{\text{a}}\vec{\text{b}}=3600$
$|\vec{\text{a}}|^2+\big|\vec{\text{b}}\big|^2+2\vec{\text{a}}\vec{\text{b}}=3600\dots(1)$
Now, $\big|\vec{\text{a}}-\vec{\text{b}}\big|=40$
Squaring both the sides,
$\big|\vec{\text{a}}-\vec{\text{b}}\big|^2=(40)^2$
$|\vec{\text{a}}|^2+\big|\vec{\text{b}}\big|^2-2\vec{\text{a}}\vec{\text{b}}=1600\dots(2)$
Adding (1) and (2),
$2|\vec{\text{a}}|^2+2\big|\vec{\text{b}}\big|^2+2\vec{\text{a}}\vec{\text{b}}-2\vec{\text{a}}\vec{\text{b}}=3600-1600$
$2|\vec{\text{a}}|^2+2(46)^2=5200$
$2|\vec{\text{a}}|^2=5200-4232$
$2|\vec{\text{a}}|^2=968$
$|\vec{\text{a}}|^2=\frac{968}{2}$
$|\vec{\text{a}}|^2=484$
$|\vec{\text{a}}|=\sqrt{484}$
$|\vec{\text{a}}|=22$

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 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

Determine the binomial distribution whose mean is 20 and variance 16.
→
Find the vector equation of the line passing through $(1, 2, 3)$ and parallel to the planes $\vec{\text{r}}\cdot(\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}})=5$ and $\vec{\text{r}}\cdot(3\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})=6.$
→
Evaluate the following integrals:
$\int(\text{e}^{\log\text{x}}+\sin\text{x})\cos\text{x dx}$
→
$\int\frac{\text{x}^2}{\sqrt{1-\text{x}}}\text{dx}$
→
Evaluate the definite integral in Exercise:
$\int^{4}_{1}[\text{x}-1|+|\text{x}-2|+|\text{x}-3|]\text{dx}$
→
Consider $f :\{1, 2, 3\} \rightarrow \{a, b, c\}$ and $g : \{a, b, c\} \rightarrow \{$apple, ball, cat$\} $defined as $f(1) = a, f(2) = b, f(3) = c, g(a) =$ apple, $g(b) =$ ball and $g(c) =$ cat. Show that $f, g$ and gof are invertible. Find $f^{-1}, g^{-1}$ and $gof^{-1}$ and show that $(gof)^{-1} = f^{-1}og^{-1}.$
→
Evaluate the following intregals:
$\int\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}\ \text{dx}$
→
Using integration, find the area of the region in the first quadrant enclosed by the $Y-$ axis, the line $y = x$ and the circle $x^2+y^2=32$
→
A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer’s profit on an item of model A is Rs. 15 and on an item of model B is Rs. 10. How many of items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.
→
Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(3\hat{\text{i}}+5\hat{\text{j}}+7\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}\big)$ and $\vec{\text{r}}=-\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu\big(7\hat{\text{i}}-6\hat{\text{j}}+\hat{\text{k}}\big)$
→