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Maths-

General

Easy

Question

The distance of the point $3 i with ˆ on top plus 5 k with ˆ on top$ from the line parallel to vector $6 i with ˆ on top plus j with ˆ on top minus 2 k with ˆ on top$ and passing through the point $8 i with ˆ on top plus 3 j with ˆ on top plus k with ˆ on top$ is

2 units
3 units
4 units
5 units

hint Hint:

We are given the position vector of a point. We are given a parallel vector to a line and the point through which the lines is passing. We have to find the distance between the point and the line.

The correct answer is: 3 units

We have to find the distance between a point and a line.
The equation of any line is given as follows:
$r with rightwards arrow on top equals a with rightwards arrow on top space plus lambda b with rightwards arrow on top H e r e comma space a with rightwards arrow on top space i s space t h e space p o s t i o n space v e c t o r space o f space a space p o i n t space t h r o u g h space w h i c h space t h e space l i n e space i s space p a s sin g. space space space space space space space space space space space b with rightwards arrow on top space i s space t h e space v e c t o r space p a r a l l e l space t o space t h e space l i n e space space space space space space space space space space space lambda space i s space a space c o n s tan t.$ $F o r space t h i s space q u e s t i o n comma space a with rightwards arrow on top space equals space 8 i with hat on top space plus space 3 j with hat on top space plus space k with hat on top space space a n d space b with rightwards arrow on top space equals space 6 i with hat on top space plus stack j space with hat on top minus 2 k with hat on top$ $S o comma space r with rightwards arrow on top equals 8 i with hat on top space plus space 3 j with hat on top space plus space k with hat on top space plus space lambda left parenthesis 6 i with hat on top space plus space j with hat on top space minus 2 k with hat on top right parenthesis$ $L e t space t h e space p o i n t space b e space d e n o t e d space b y space c with rightwards arrow on top equals 3 i with hat on top space plus space 5 k with hat on top$
We will find the distance between the line and point.
$N o w comma space t h e space f o r m u l a space o f space p o i n t space f r o m space t h e space l i n e space i s space g i v e n space a s space f o l l o w s colon D i s tan c e space equals space fraction numerator open vertical bar left parenthesis c with rightwards arrow on top minus a with rightwards arrow on top right parenthesis space cross times space b with rightwards arrow on top close vertical bar over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction W e space w i l l space f i n d space t h e space v a l u e s space o n e space b y space o n e. c with rightwards arrow on top space minus space a with rightwards arrow on top space equals space left parenthesis 3 space minus space 8 right parenthesis i with hat on top space plus space left parenthesis 0 space minus space 3 right parenthesis j with hat on top space plus space left parenthesis 5 space minus space 1 right parenthesis k with hat on top space space space space space space space space space space space space equals space minus 5 i with hat on top space minus 3 j with hat on top space plus 4 k with hat on top space space space space space space space space space space space space space space space space space space blank subscript blank space space space space space space space space space space space space space space space space space$
$open vertical bar b with rightwards arrow on top close vertical bar equals square root of 6 squared space plus space 1 squared space plus space 2 squared space end root space space space space space space equals square root of 36 space plus space 1 plus space 4 end root space space space space space space equals square root of 41 space space space space space space$
Now, we will find the cross product.
$left parenthesis c with rightwards arrow on top space minus space a with rightwards arrow on top right parenthesis space cross times space b with rightwards arrow on top space equals space open vertical bar table row cell i with hat on top end cell cell j with hat on top end cell cell k with bar on top end cell row cell negative 5 end cell cell negative 3 end cell 4 row 6 1 cell negative 2 end cell end table close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space equals left parenthesis 6 space minus space 4 right parenthesis i with hat on top space minus space left parenthesis 10 space minus space 24 right parenthesis j with hat on top plus left parenthesis negative 5 space plus space 18 right parenthesis k with hat on top space space space space space space space space space space space space space space space space space space space space space space space space equals space 2 i with hat on top space plus space 14 stack j space with hat on top space plus space 13 k with hat on top$
We will resubstitute all the values.
$D i s tan c e space equals fraction numerator open vertical bar 2 i with hat on top space plus 14 j with hat on top space plus 13 k with hat on top space close vertical bar over denominator square root of 41 end fraction space space space space space space space space space space space space space space space space space equals fraction numerator square root of 2 squared space plus space 14 squared space plus space 13 squared end root over denominator square root of 41 end fraction space space space space space space space space space space space space space space space space space equals fraction numerator square root of 4 space plus space 196 space plus space 169 end root over denominator square root of 41 end fraction space space space space space space space space space space space space space space space space space equals square root of 369 over 41 end root space space space space space space space space space space space space space space space space space equals square root of 9 space space space space space space space space space space space space space space space space space equals space 3 space u n i t s$
So, the distance between the line and the point is 3 units.

For such questions, we should know different formulas of distances for line or plane.

Related Questions to study

The value of 'a' so that the volume of parallel piped formed by $i with ˆ on top plus a j with ˆ on top plus k with ˆ on top comma j with ˆ on top plus a k with ˆ on top$ and $a with ˆ on top plus k with ˆ on top$ becomes minimum is

For such questions, we should know the volume of parallelepiped.

The value of 'a' so that the volume of parallel piped formed by $i with ˆ on top plus a j with ˆ on top plus k with ˆ on top comma j with ˆ on top plus a k with ˆ on top$ and $a with ˆ on top plus k with ˆ on top$ becomes minimum is

For such questions, we should know the volume of parallelepiped.

There are following extraction process of silver but not

There are following extraction process of silver but not

Chemistry-General

$I n space triangle A B C comma space i f space x equals tan invisible function application left parenthesis left parenthesis B minus C right parenthesis divided by 2 right parenthesis tan invisible function application A divided by 2 comma y equals tan invisible function application left parenthesis left parenthesis C minus A right parenthesis divided by 2 right parenthesis tan invisible function application B divided by 2 comma z equals tan invisible function application left parenthesis left parenthesis A minus B right parenthesis divided by 2 right parenthesis tan invisible function application C divided by 2 space t space h space e n space x plus y plus z space$
in terms terms only is

$I n space triangle A B C comma space i f space x equals tan invisible function application left parenthesis left parenthesis B minus C right parenthesis divided by 2 right parenthesis tan invisible function application A divided by 2 comma y equals tan invisible function application left parenthesis left parenthesis C minus A right parenthesis divided by 2 right parenthesis tan invisible function application B divided by 2 comma z equals tan invisible function application left parenthesis left parenthesis A minus B right parenthesis divided by 2 right parenthesis tan invisible function application C divided by 2 space t space h space e n space x plus y plus z space$
in terms terms only is

parallel

$n space triangle A B C space left parenthesis c o t invisible function application A divided by 2 plus c o t invisible function application B divided by 2 plus c o t invisible function application C divided by 2 right parenthesis divided by left parenthesis c o t invisible function application A plus c o t invisible function application B plus c o t invisible function application C right parenthesis equals$

$n space triangle A B C space left parenthesis c o t invisible function application A divided by 2 plus c o t invisible function application B divided by 2 plus c o t invisible function application C divided by 2 right parenthesis divided by left parenthesis c o t invisible function application A plus c o t invisible function application B plus c o t invisible function application C right parenthesis equals$

$I n space triangle A B C comma space i f space x equals$ $tan invisible function application left parenthesis left parenthesis B minus C right parenthesis divided by 2 right parenthesis tan invisible function application A divided by 2 comma y equals tan invisible function application left parenthesis left parenthesis C minus A right parenthesis divided by 2 right parenthesis tan invisible function application B divided by 2 comma z equals tan invisible function application left parenthesis left parenthesis A minus B right parenthesis divided by 2 right parenthesis tan invisible function application C divided by 2 space t space h space e n space x plus y plus z space i n space t e r m s space left parenthesis x comma y comma z space t e r m s space o n l y space right parenthesis space i s$

$I n space triangle A B C comma space i f space x equals$ $tan invisible function application left parenthesis left parenthesis B minus C right parenthesis divided by 2 right parenthesis tan invisible function application A divided by 2 comma y equals tan invisible function application left parenthesis left parenthesis C minus A right parenthesis divided by 2 right parenthesis tan invisible function application B divided by 2 comma z equals tan invisible function application left parenthesis left parenthesis A minus B right parenthesis divided by 2 right parenthesis tan invisible function application C divided by 2 space t space h space e n space x plus y plus z space i n space t e r m s space left parenthesis x comma y comma z space t e r m s space o n l y space right parenthesis space i s$

Two sides of triangle are given by equation and angle between
the sides is $pi divided by 3.$ Then perimeter of the triangle is

Two sides of triangle are given by equation and angle between
the sides is $pi divided by 3.$ Then perimeter of the triangle is

parallel

$I n a triangle A B C comma a left parenthesis c o s squared invisible function application B plus c o s squared invisible function application C right parenthesis plus c o s invisible function application A left parenthesis c c o s invisible function application C plus b c o s invisible function application B right parenthesis equals$

$I n a triangle A B C comma a left parenthesis c o s squared invisible function application B plus c o s squared invisible function application C right parenthesis plus c o s invisible function application A left parenthesis c c o s invisible function application C plus b c o s invisible function application B right parenthesis equals$

$left parenthesis I n space triangle A B C comma left parenthesis cos invisible function application C plus cos invisible function application A right parenthesis divided by left parenthesis c plus a right parenthesis plus left parenthesis cos invisible function application B right parenthesis divided by b equals & right parenthesis$

$left parenthesis I n space triangle A B C comma left parenthesis cos invisible function application C plus cos invisible function application A right parenthesis divided by left parenthesis c plus a right parenthesis plus left parenthesis cos invisible function application B right parenthesis divided by b equals & right parenthesis$

$I n space capital delta A B C sum a cubed c o s invisible function application left parenthesis B minus C right parenthesis equals$

$I n space capital delta A B C sum a cubed c o s invisible function application left parenthesis B minus C right parenthesis equals$

parallel

$I n space a n y space triangle A B C left parenthesis left parenthesis a plus b plus c right parenthesis left parenthesis b plus c minus a right parenthesis left parenthesis c plus a minus b right parenthesis left parenthesis a plus b minus c right parenthesis right parenthesis divided by left parenthesis 4 b squared c squared right parenthesis$

$I n space a n y space triangle A B C left parenthesis left parenthesis a plus b plus c right parenthesis left parenthesis b plus c minus a right parenthesis left parenthesis c plus a minus b right parenthesis left parenthesis a plus b minus c right parenthesis right parenthesis divided by left parenthesis 4 b squared c squared right parenthesis$

$142 I n triangle A B C sum a cubed S i n invisible function application left parenthesis B minus C right parenthesis equals$

$142 I n triangle A B C sum a cubed S i n invisible function application left parenthesis B minus C right parenthesis equals$

$I n space triangle A B C comma sum a left parenthesis sin invisible function application B minus sin invisible function application C right parenthesis equals$

$I n space triangle A B C comma sum a left parenthesis sin invisible function application B minus sin invisible function application C right parenthesis equals$

parallel

The perimeter of a Δ $A B C space$ is 6 times the A.M of the sines of its angles if the side ' is , then the angle is

The perimeter of a Δ $A B C space$ is 6 times the A.M of the sines of its angles if the side ' is , then the angle is

The angles of a triangle are in the ratio 3:5:10. Then ratio of smallest to greatest side is

The angles of a triangle are in the ratio 3:5:10. Then ratio of smallest to greatest side is

If the angles of triangle are in the ratio 1:1:4 then ratio of the perimeter of triangle to its largest side

If the angles of triangle are in the ratio 1:1:4 then ratio of the perimeter of triangle to its largest side

parallel

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