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Maths-

General

Easy

Question

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

$\frac{[a b c]}{| a \times b + c \times a + b \times c |}$
$\frac{2 [a b c]}{| a \times b + b \times c + c \times a |}$
$[a b c]$
None of these

The correct answer is: $fraction numerator left square bracket a b c right square bracket over denominator vertical line a cross times b plus c cross times a plus b cross times c vertical line end fraction$

The vector equation of the plane passing through points $a comma b comma c$ is $r. left parenthesis a cross times b plus b cross times c plus c cross times a right parenthesis equals left square bracket a b c right square bracket$
Therefore, the length of the perpendicular from the origin to this plane is given by $fraction numerator left square bracket a b c right square bracket over denominator vertical line a cross times b plus b cross times c plus c cross times a vertical line end fraction$ .

Related Questions to study

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

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

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

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

parallel

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:

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

parallel

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

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

parallel

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

The point to which the origin should be shifted in order to eliminate x and y terms in the equation $4 x to the power of 2 end exponent plus 9 y to the power of 2 end exponent minus 8 x plus 36 y plus 4$ = 0

The point to which the origin should be shifted in order to eliminate x and y terms in the equation $4 x to the power of 2 end exponent plus 9 y to the power of 2 end exponent minus 8 x plus 36 y plus 4$ = 0

parallel

Orthocentre of the triangle is (2,1) and circumcentre is $open parentheses fraction numerator text 7 end text over denominator text 2 end text end fraction text , end text fraction numerator text 5 end text over denominator text 2 end text end fraction close parentheses$ , then its nine point circle centre is

Orthocentre of the triangle is (2,1) and circumcentre is $open parentheses fraction numerator text 7 end text over denominator text 2 end text end fraction text , end text fraction numerator text 5 end text over denominator text 2 end text end fraction close parentheses$ , then its nine point circle centre is

If a vertex of a $capital delta to the power of l e end exponent$ is (1,1) and the midpoints of two sides through this vertex are (-1,2)(3,2) then centroid of the triangle is

If a vertex of a $capital delta to the power of l e end exponent$ is (1,1) and the midpoints of two sides through this vertex are (-1,2)(3,2) then centroid of the triangle is

$I f blank A equals open parentheses 1 comma negative 1 close parentheses comma B equals open parentheses 2 comma 1 close parentheses comma C equals open parentheses 3 comma 0 close parentheses comma blank t h e n blank t h e blank c i r c u m minus r a d i u s blank o f blank increment A B C blank i s$

$I f blank A equals open parentheses 1 comma negative 1 close parentheses comma B equals open parentheses 2 comma 1 close parentheses comma C equals open parentheses 3 comma 0 close parentheses comma blank t h e n blank t h e blank c i r c u m minus r a d i u s blank o f blank increment A B C blank i s$

parallel

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