00:02
Okay, so we are given a point a the point a is three comma negative one comma five and the equation of line l in the vector form is given as r is equal to eight i cap plus zero k k cap sorry zero j cap eight i cap plus zero j cap minus k cap plus some parameter t times minus six i cap plus j cap plus 4 k cap so we need to find a point b which lies on the line l such that ab is perpendicular to l so any point on the line given by this l is of the form 8 minus 60 0 plus t that is t and minus 1 plus 4 t now the ab vector will be how to find the vector a b the position vector of b minus position vector of a that means you subtract the coordinates 8 minus 60 minus 3 which is 5 minus 60 i cap and plus t plus 1 j cap and minus 1 plus 4 t minus 5 that is 4 t minus 6 k cap but this is perpendicular to the line what is the direction of the line this is the direction vector of the line so the direction vector of the line d l is equal to minus 6i plus j plus 4k these two are perpendicular so that means the dot product should be 0 so what is the dot product so negative 6 into 5 minus 60 plus component of j is a one year and you have t plus 1 so 1 times of t plus 1 plus 4 times of 40 minus 6 that should be 0 so let's simplify this equation so that means you get minus 30 plus 36 t plus t plus 1 plus 16 t minus 24 is equal to 0.
02:13
So 36 t plus 37 t is 37 plus 67 plus 16 is 53 t and minus 29 plus minus 24 in minus 53 is equal to 0.
02:24
So t is equal 1.
02:25
That t equal 1 you replace here.
02:28
So the coordinates of b will be 2 comma 1 comma 3.
02:32
So, this is the point b.
02:35
So, the answer for coordinates of b is 2 comma, 1, 3.
02:39
Because t value is one, i substituted that.
02:43
And the next question that was asked is, given a plane pi, so there is a plain pi, are dot, i minus j plus 3k is equal to 15.
03:03
Find the coordinates of c where the line l intersects the plane pi.
03:07
So the equation of the plane in cartesian form is x minus y plus tz is equal to 50.
03:15
Now we need to find if this is a plane pi that line l is this, so bad das, this line meets the plane.
03:25
So we already know any point in the line is of the form 8 minus 60 t minus 1 plus 40.
03:30
So let's substitute 8 minus 60 in place of x minus y which is t plus 3 times of negative 1 plus 40.
03:40
This is equal to 15 so let's simplify this equation so it is 8 minus 60 minus t minus 3 plus 12 t is equal to 15 for 5 plus 5 t is equal to 15 or 5 t is equal to 10 t is equal to 2 so that means the point where the line meets the plane is substitute t equal to in the general form of the line that is 8 minus 60 t minus 1 plus 40 so we call that as a point c so 8 minus 6, 8 minus 6 into 2, 2, 2, negative 1 plus 40...