Please answer this question showing all steps let u 2 7 3...

Please answer this question showing all steps: Let u = ⟨2, 7, −3⟩. Determine whether u is parallel or perpendicular to the following vectors. If it is neither, compute the area of the parallelogram with sides given by this vector and u. a) v = ⟨1, 5, 9⟩ b) w = i + j + 3k c) s = ⟨−6, −21, 9⟩ d) a unit vector normal to the plane 2x + y − 6z = 7.

Transcript

00:01 Let's understand this problem diagramically.

00:03 So here we have two planes.

00:05 The first plane i have shown in black color, that is 4x minus 3y plus 2z equals 12.

00:11 This is one plane.

00:13 And there is another plane which i have shown in green color.

00:17 That is x plus yy minus z equals 7.

00:20 And these two planes intersect.

00:22 And when these two planes intersect, we see that there is a line form.

00:27 So this is basically the line of intersection.

00:30 So we can determine the equation of this line.

00:33 And then we have to determine the equation of the plane which i have shown in blue color.

00:38 And this plane is perpendicular to the line of intersection.

00:42 So basically we can consider that the line of intersection as a sort of normal to this plane.

00:48 So we have a point on the plane that is 4 .3 comma minus 2, which lies on the plane as well as a normal vector, which is along the direction of the line of intersection.

01:00 So once we have these two information, that is the point as well as a normal vector to the plane, we can find the equation of the plane.

01:09 So our first task is to determine the equation of the line of intersection.

01:14 That is, we have to find the equation of the line formed when these two planes intersect.

01:20 So let's see how to find the line which is formed by the intersection of these two planes.

01:26 To determine the equation of the line, i'm going to use the equation of the.

01:31 The prints because the line is formed by the intersection of these two planes.

01:36 So we have the first equation that is the one of the plane is 4x minus 3y plus 2 z and this equals 12 and the other equation is x plus piy minus z equals 7.

01:53 So we have to solve these two equations of plane so that we get the equation of line and the equation of line that we are going to form we will we will form in parametric form.

02:06 It is we will get in parametric form.

02:13 So we can solve this by elimination method.

02:17 I'm going to eliminate z, which means i can multiply this equation by 2 and add with the equation 1.

02:25 So i'm going to write the first equation as it is.

02:29 It is 4x minus 3y plus 2z.

02:32 This equals 12.

02:34 And down below that, i add the.

02:37 The second equation by multiplying by 2.

02:41 So i have to multiply each terms by 2.

02:43 This becomes 2x.

02:45 5y times of 2 is 10y.

02:49 And then negative z times of 2 is negative 2 z.