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15. The Cross Product: Suppose that u = <u1, u2, u3> and v = <v1, v2, v3> ? R^3. Let us define the vector: u x v = (u2v3 - u3v2)i - (u1v3 - u3v1)j + (u1v2 - u2v1)k, called the cross product of u and v. Prove that for any u, v ? R^3: u x v is orthogonal to both u and v. 16. Intersecting Lines: Consider the two lines given by: <x, y, z> = <3 - 5t, 7 + 2t, 4 - t>, and <x, y, z> = <7 + 3s, 5 - s, 10 - 2s>. a. Show that the two lines intersect at exactly one point, and find the coordinates of that point. b. Find a Cartesian equation for the plane that contains the two lines. Hint: use the cross-product on the two direction vectors. 17. Orthogonal Lines: We will say that two lines L1 and L2 are orthogonal to each other if they intersect and any direction vector for L1 is orthogonal to any direction vector for L2. Show that the lines <x, y, z> = <3 - t, 5t, -7 + 4t>, and <x, y, z> = <8 + 3s, 3 - s, 1 + 2s> are orthogonal to each other, and find a Cartesian equation for the plane containing both lines. Where do the lines intersect?

          15. The Cross Product: Suppose that u = <u1, u2, u3> and v = <v1, v2, v3> ? R^3.
Let us define the vector:
u x v = (u2v3 - u3v2)i - (u1v3 - u3v1)j + (u1v2 - u2v1)k,
called the cross product of u and v.
Prove that for any u, v ? R^3: u x v is orthogonal to both u and v.
16. Intersecting Lines: Consider the two lines given by:
<x, y, z> = <3 - 5t, 7 + 2t, 4 - t>, and <x, y, z> = <7 + 3s, 5 - s, 10 - 2s>.
a. Show that the two lines intersect at exactly one point, and find the coordinates of that point.
b. Find a Cartesian equation for the plane that contains the two lines. Hint: use the cross-product on the two direction vectors.
17. Orthogonal Lines: We will say that two lines L1 and L2 are orthogonal to each other if they intersect and any direction vector for L1 is orthogonal to any direction vector for L2.
Show that the lines <x, y, z> = <3 - t, 5t, -7 + 4t>, and <x, y, z> = <8 + 3s, 3 - s, 1 + 2s> are orthogonal to each other, and find a Cartesian equation for the plane containing both lines. Where do the lines intersect?

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15. The Cross Product: Suppose that u = <u1, u2, u3> and v = <v1, v2, v3> ? R^3.
Let us define the vector:
u x v = (u2v3 - u3v2)i - (u1v3 - u3v1)j + (u1v2 - u2v1)k,
called the cross product of u and v.
Prove that for any u, v ? R^3: u x v is orthogonal to both u and v.
16. Intersecting Lines: Consider the two lines given by:
<x, y, z> = <3 - 5t, 7 + 2t, 4 - t>, and <x, y, z> = <7 + 3s, 5 - s, 10 - 2s>.
a. Show that the two lines intersect at exactly one point, and find the coordinates of that point.
b. Find a Cartesian equation for the plane that contains the two lines. Hint: use the cross-product on the two direction vectors.
17. Orthogonal Lines: We will say that two lines L1 and L2 are orthogonal to each other if they intersect and any direction vector for L1 is orthogonal to any direction vector for L2.
Show that the lines <x, y, z> = <3 - t, 5t, -7 + 4t>, and <x, y, z> = <8 + 3s, 3 - s, 1 + 2s> are orthogonal to each other, and find a Cartesian equation for the plane containing both lines. Where do the lines intersect?

Added by Brianna J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

07/27/2023

Step 1

To do this, we need to show that the dot product of the cross product with u and v is zero. Let u = (u1, u2, u3) and v = (v1, v2, v3). Then, the cross product of u and v is given by: u x v = (u2v3 - u3v2)i - (u1v3 - u3v1)j + (u1v2 - u2v1)k Now, let's compute Show more…

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15. The Cross Product: Suppose that u = <u1, u2, u3> and v = <v1, v2, v3> ∈ R^3. Let us define the vector: u x v = (u2v3 – u3v2)i – (u1v3 – u3v1)j + (u1v2 – u2v1)k, called the cross product of u and v. Prove that for any u, v ∈ R^3: u x v is orthogonal to both u and v. 16. Intersecting Lines: Consider the two lines given by: <x, y, z> = <3 – 5t, 7 + 2t, 4 – t>, and <x, y, z> = <7 + 3s, 5 – s, 10 – 2s>. a. Show that the two lines intersect at exactly one point, and find the coordinates of that point. b. Find a Cartesian equation for the plane that contains the two lines. Hint: use the cross-product on the two direction vectors. 17. Orthogonal Lines: We will say that two lines L1 and L2 are orthogonal to each other if they intersect and any direction vector for L1 is orthogonal to any direction vector for L2. Show that the lines <x, y, z> = <3 – t, 5t, –7 + 4t>, and <x, y, z> = <8 + 3s, 3 – s, 1 + 2s> are orthogonal to each other, and find a Cartesian equation for the plane containing both lines. Where do the lines intersect?

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Transcript

00:01 Hello students, first we have to find u x v .u equals to 0.

00:06 So, u x v .u equals to u2v3 minus u3v2 into u1 plus u3v1 minus u1v3 into u2 plus u3 into u1v2 minus u2v1.

00:31 So, expanding the dot product, we get u1u2v3 minus u1u3v2 plus u2u3v1 minus u2u1v3 plus u3u1v2 minus u2v1u3.

00:57 Notice that the terms in the dot product cancel each other out.

01:02 So, u x v .u equals to 0.

01:06 Therefore, u x v is orthogonal to u.

01:16 Similarly, we can see that u x v .v is equals to 0.

01:26 So, we can say that u x v is orthogonal to vector view.

01:35 Now, the next part is to show that the line, the two lines intersect at exactly one point.

01:44 We need to find the values of t and s for which the two lines have the same point of intersection.

01:50 So, set the coordinates of the two lines equal to each other.

01:55 For the x coordinate, 3 minus 5t equals to 7 plus 3s.

02:00 For the y coordinate, 7 plus 2t equals to 5 minus s.

02:07 For the z coordinate, 4 minus t equals to 10 minus 2s.

02:12 Now, we solve the system of equation.

02:14 So, this implies s equals to 5t minus 4 by 3.

02:21 This implies t equals to minus 10 by 11.

02:29 Now, substitute the value into t, s equals to minus 94 by 33.

02:39 Now, we have to find the points of intersection by plugging the values of t and s.

02:47 So, using the first equation to find the x coordinate, x equals to 3 minus 5t...

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