Find direction numbers for the line of intersection of the planes x+y+z=1 and x+z=0

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Todd Williams · Accepted Answer

Note that the line of intersection of two planes must be parallel to both planes. The cross product of the two planes

Nick Camelot · Accepted Answer

Result:⟨−1,2,1⟩Solution:The given equations are:x+y+z=1(Plane&amp;nbsp;1)x+z=0(Plane&amp;nbsp;2)To find the direction numbers, we can find the vector that is orthogonal to both planes. This vector will be parallel to the line of intersection.Let&#039;s solve the system of equations:We can rewrite Plane 1 as:x+y+z=1⇒y=1−x−zSubstituting this value of y into Plane 2:x+z=0Solving for x, we get:x=−zSubstituting this value of x into Plane 1:y=1−(−z)−z=1+2zSo, we have the parameterization of the line of intersection as:x=−zy=1+2zz=zTherefore, the direction numbers of the line of intersection are {−1,2,1}.In vector form, the direction vector is given by ⟨−1,2,1⟩.

Mr Solver · Accepted Answer

Let&#039;s consider the normal vectors N1 and N2 for the given planes. The normal vector of a plane is a vector perpendicular to the plane&#039;s surface.For the plane x+y+z=1, we can easily see that the normal vector N1 is (111).For the plane x+z=0, we can rewrite it as x+0y+z=0. The coefficients of x, y, and z in this equation give us the components of the normal vector N2, which is (101).Now, to find the direction numbers of the line of intersection, we can take the cross product of the normal vectors:𝐝=N1×N2Let&#039;s calculate the cross product:𝐝=(111)×(101)Using the determinant formula for cross products, we have:𝐝=(1−11)Therefore, the direction numbers for the line of intersection are (1−11).

madeleinejames20 · Accepted Answer

Step 1:To find the direction numbers for the line of intersection of the planes x+y+z=1 and x+z=0, we can first express both planes in the general form of an equation. Then, by comparing the coefficients, we can determine the direction numbers.Let&#039;s begin by writing the equation of the first plane in the general form:x+y+z=1Next, we&#039;ll express the equation of the second plane in the same form:x+0y+z=0Step 2:Now, we can compare the coefficients of x, y, and z in both equations to determine the direction numbers.From the first plane, we have:Coefficient of x: 1Coefficient of y: 1Coefficient of z: 1From the second plane, we have:Coefficient of x: 1Coefficient of y: 0Coefficient of z: 1Therefore, the direction numbers for the line of intersection are (1,1,1) and (1,0,1).Step 3:We can also write the direction vector in parametric form as:𝐯=(111)×(101)By taking the cross product of the direction vectors, we get the direction vector 𝐯.Note: The cross product × is used to calculate the vector product.Thus, the direction vector for the line of intersection is given by 𝐯=(1−11).

Find direction numbers for the line of intersection of the

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