## Linear Algebra and Its Applications, Review Exercise 1.24

Review exercise 1.24. The equation $u + 2v - w = 6$ defines a plane in 3-space. Find equations that define the following:

(a) a plane parallel to the first plane but going through the origin

(b) a second plane that (like the original plane) contains the points $(6, 0, 0)$ and $(2, 2, 0)$

(c) a third plane that intersects the original plane and the one from (b) in the point $(4, 1, 0)$

Answer: (a) A plane passing through the origin must correspond to an equation that holds true for $u = v = w = 0$. The equation $u + 2v - w = 0$ satisfies this condition, and produces a plane parallel to the original plane.

(b) The points $(6, 0, 0)$ and $(2, 2, 0)$ satisfy the original equation $u + 2v - w = 6$ and are in the plane defined by that equation. For both these points we have $w = 0$. Therefore if we take the equation for the original plane and change the term involving $w$ we can produce a new and different equation corresponding to a new plane containing these points. One such equation is $u + 2v + w = 6$.

(c) For the point $(4, 1, 0)$ we have $w = 0$, and so as in (b) the equation we produce may have any term involving $w$. We simply need to ensure that the equation is satisfied when $u = 4$ and $v = 1$. One such equation is $2u - 2v + w = 6$. Since the point $(4, 1, 0)$ satisfies both this equation, the original equation, and the equation from (b) above, it is at the intersection of all three planes.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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