Have a query? Ask a Tutor!!

Access to best educators and exclusive paper discussions

Offsite Campus Turito Championship

Can’t find what you’re looking for?

Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

Homework- One to One Solutions
Understand concepts to solve better
Get your doubts solved instantly

Maths-

General

Easy

Question

If $u equals 2 i plus 2 j minus k$ and $v equals 6 i minus 3 j plus 2 k comma$ then a unit vector perpendicular to both u and v is

$i - 10 j - 18 k$
$\frac{1}{\sqrt{17}} (\frac{1}{5} i - 2 j - \frac{18}{5} k)$
$\frac{1}{\sqrt{473}} (7 i - 10 j - 18 k)$
None of these

The correct answer is: $fraction numerator 1 over denominator square root of 17 end fraction open parentheses fraction numerator 1 over denominator 5 end fraction i minus 2 j minus fraction numerator 18 over denominator 5 end fraction k close parentheses$

$u equals 2 i plus 2 j minus k$ and $v equals 6 i minus 3 j plus 2 k$
Let vector $c equals c subscript 1 end subscript i plus c subscript 2 end subscript j plus c subscript 3 end subscript k$ is perpendicular to both $u$ and $v comma$ then $c. u equals 0$
$rightwards double arrow 2 c subscript 1 end subscript plus 2 c subscript 2 end subscript minus c subscript 3 end subscript equals 0$ …..(i)
and $c. v equals 0 rightwards double arrow 6 c subscript 1 end subscript minus 3 c subscript 2 end subscript plus 2 c subscript 3 end subscript equals 0$ …..(ii)
Solving equation (i) and (ii) by cross multiplication
$fraction numerator c subscript 1 end subscript over denominator 4 minus 3 end fraction equals fraction numerator c subscript 2 end subscript over denominator negative 6 minus 4 end fraction equals fraction numerator c subscript 3 end subscript over denominator negative 6 minus 12 end fraction equals lambda$ , (say)
$rightwards double arrow fraction numerator c subscript 1 end subscript over denominator 1 end fraction equals fraction numerator c subscript 2 end subscript over denominator negative 10 end fraction equals fraction numerator c subscript 3 end subscript over denominator negative 18 end fraction equals lambda$
$rightwards double arrow c subscript 1 end subscript equals lambda comma$ $c subscript 2 end subscript equals negative 10 lambda$ and $c subscript 3 end subscript equals negative 18 lambda$
Thus $c equals lambda left parenthesis i minus 10 j minus 18 k right parenthesis$
$vertical line c vertical line equals lambda square root of 1 plus 100 plus 324 end root equals lambda square root of 425$
Hence required unit vector is, $fraction numerator c over denominator vertical line c vertical line end fraction$
$equals fraction numerator lambda left parenthesis i minus 10 j minus 18 k right parenthesis over denominator lambda square root of 425 end fraction equals fraction numerator 1 over denominator square root of 425 end fraction left parenthesis i minus 10 j minus 18 k right parenthesis$
= $fraction numerator 1 over denominator 5 square root of 17 end fraction left parenthesis i minus 10 j minus 18 k right parenthesis equals fraction numerator 1 over denominator square root of 17 end fraction open parentheses fraction numerator 1 over denominator 5 end fraction i minus 2 j minus fraction numerator 18 over denominator 5 end fraction k close parentheses$
Aliter : Required vector is $fraction numerator u cross times v over denominator vertical line u cross times v vertical line end fraction equals fraction numerator i minus 10 j minus 18 k over denominator square root of 425 end fraction.$

Related Questions to study

The equation of plane passing through a point $A left parenthesis 2 comma negative 1 , 3 right parenthesis$ and parallel to the vectors $a equals left parenthesis 3 , 0 comma negative 1 right parenthesis$ and $b equals left parenthesis negative 3 , 2 comma 2 right parenthesis$ is

The equation of plane passing through a point $A left parenthesis 2 comma negative 1 , 3 right parenthesis$ and parallel to the vectors $a equals left parenthesis 3 , 0 comma negative 1 right parenthesis$ and $b equals left parenthesis negative 3 , 2 comma 2 right parenthesis$ is

The equation of the plane passing through the points $left parenthesis negative 1 comma negative 2 , 0 right parenthesis comma left parenthesis 2 , 3 comma 5 right parenthesis$ and parallel to the line
$r equals negative 3 j plus k plus lambda left parenthesis 2 i plus 5 j minus k right parenthesis$ is

The equation of the plane passing through the points $left parenthesis negative 1 comma negative 2 , 0 right parenthesis comma left parenthesis 2 , 3 comma 5 right parenthesis$ and parallel to the line
$r equals negative 3 j plus k plus lambda left parenthesis 2 i plus 5 j minus k right parenthesis$ is

The distance from the point $negative i plus 2 j plus 6 k$ to the straight line through the point (2, 3, –4) and parallel to the vector $6 i plus 3 j minus 4 k$ is

The distance from the point $negative i plus 2 j plus 6 k$ to the straight line through the point (2, 3, –4) and parallel to the vector $6 i plus 3 j minus 4 k$ is

parallel

The length of the perpendicular from the origin to the plane passing through three non-collinear points $a comma b comma c$ is

The length of the perpendicular from the origin to the plane passing through three non-collinear points $a comma b comma c$ is

A vector $n$ of magnitude 8 units is inclined to x-axis at $4 5 to the power of o end exponent$ , y-axis at $6 0 to the power of o end exponent$ and an acute angle with z-axis. If a plane passes through a point $left parenthesis square root of 2 comma negative 1 , 1 right parenthesis$ and is normal to $n$ , then its equation in vector form is

A vector $n$ of magnitude 8 units is inclined to x-axis at $4 5 to the power of o end exponent$ , y-axis at $6 0 to the power of o end exponent$ and an acute angle with z-axis. If a plane passes through a point $left parenthesis square root of 2 comma negative 1 , 1 right parenthesis$ and is normal to $n$ , then its equation in vector form is

$fraction numerator text d end text over denominator text dx end text end fraction open square brackets s i n open parentheses l o g open parentheses s i n e to the power of x to the power of 2 end exponent end exponent close parentheses close parentheses close square brackets$ =

$fraction numerator text d end text over denominator text dx end text end fraction open square brackets s i n open parentheses l o g open parentheses s i n e to the power of x to the power of 2 end exponent end exponent close parentheses close parentheses close square brackets$ =

parallel

If $square root of T a n y end root$ = $e to the power of c o s 2 x end exponent. s i n x comma blank t h e n blank fraction numerator text dy end text over denominator text dx end text end fraction$ =

If $square root of T a n y end root$ = $e to the power of c o s 2 x end exponent. s i n x comma blank t h e n blank fraction numerator text dy end text over denominator text dx end text end fraction$ =

The line $chi plus gamma equals 2$ is tangent to tthe curve $chi to the power of 2 end exponent equals 3 minus 2 gamma$ at its point

The line $chi plus gamma equals 2$ is tangent to tthe curve $chi to the power of 2 end exponent equals 3 minus 2 gamma$ at its point

In general calcium carbonate is not soluble in water but by the addition of carbon dioxide to it, it becomes soluble. This is because:

In general calcium carbonate is not soluble in water but by the addition of carbon dioxide to it, it becomes soluble. This is because:

parallel

The correct stability order for the following species is:
I)
II)
III)
IV)

The correct stability order for the following species is:
I)
II)
III)
IV)

Chemistry-General

The point (4,3) is translated to the point (3,1) and then axes rotated through 30º about the origin then the new position of point is

The point (4,3) is translated to the point (3,1) and then axes rotated through 30º about the origin then the new position of point is

The arrangement of the following in ascending order of angle of rotation to eliminate xy term in the following equations.
i) 2 $text x end text to the power of 2 end exponent$ +5xy+2 $text y end text to the power of 2 end exponent$ +5x+7y+1=0
ii) 2 $text x end text to the power of 2 end exponent$ + $square root of text 3 end text end root$ xy+3 $text y end text to the power of 2 end exponent$ = 9
iii) 9 $x to the power of 2 end exponent$ +2 $square root of text 3 end text end root$ xy+7 $text y end text to the power of 2 end exponent$ = 10

The arrangement of the following in ascending order of angle of rotation to eliminate xy term in the following equations.
i) 2 $text x end text to the power of 2 end exponent$ +5xy+2 $text y end text to the power of 2 end exponent$ +5x+7y+1=0
ii) 2 $text x end text to the power of 2 end exponent$ + $square root of text 3 end text end root$ xy+3 $text y end text to the power of 2 end exponent$ = 9
iii) 9 $x to the power of 2 end exponent$ +2 $square root of text 3 end text end root$ xy+7 $text y end text to the power of 2 end exponent$ = 10

parallel

If the distance between the two given points is 2 units and the points are transferred by shifting the origin to (2,2) then the distance between the points in their new position is

If the distance between the two given points is 2 units and the points are transferred by shifting the origin to (2,2) then the distance between the points in their new position is

The transformed equation of $7 x to the power of 2 end exponent plus 2 square root of 3 x y plus 9 y to the power of 2 end exponent equals 8$ when the axes are rotated through an angle $fraction numerator text π end text over denominator text 3 end text end fraction$ is

The transformed equation of $7 x to the power of 2 end exponent plus 2 square root of 3 x y plus 9 y to the power of 2 end exponent equals 8$ when the axes are rotated through an angle $fraction numerator text π end text over denominator text 3 end text end fraction$ is

The transformed equation of $x to the power of 2 end exponent minus y to the power of 2 end exponent plus 2 x plus 4 y equals 0$ when the origin is shifted to the point (-1,2) is

The transformed equation of $x to the power of 2 end exponent minus y to the power of 2 end exponent plus 2 x plus 4 y equals 0$ when the origin is shifted to the point (-1,2) is

parallel

card img

With Turito Academy.

card img

With Turito Foundation.

card img

Get an Expert Advice From Turito.

Turito Academy

card img

With Turito Academy.

Test Prep

card img

With Turito Foundation.