# Find a set of vectors that spans the plane P1: 2x + 3y -z = 0 Question
Vectors and spaces Find a set of vectors that spans the plane $$\displaystyle{P}{1}:{2}{x}+{3}{y}-{z}={0}$$ 2021-01-06
Note that if $$\displaystyle{\left({x},{y},{z}\right)}∈{P}{1}$$ then $$\displaystyle{2}{x}+{3}{y}−{z}={0}.$$ Therefore we have $$\displaystyle{z}={2}{x}+{3}{y}$$
$$\displaystyle{\left({x},{y},{z}\right)}={\left({x},{y},{2}{x}+{3}{y}\right)}={\left({1},{0},{2}\right)}{x}+{\left({0},{1},{3}\right)}{y}∈{S}{p}{a}{n}{\left\lbrace{\left({1},{0},{2}\right)},{\left({0},{1},{3}\right)}\right\rbrace}$$
It follows that the vectors (1,0,2), (0,1,3) spans the plane P1.

### Relevant Questions Given the vector $$r(t) = { cosT, sinT, ln (CosT) }$$ and point (1, 0, 0) find vectors T, N and B at that point. $$Vector T is the unit tangent vector, so the derivative r(t) is needed. \( Vector N is the normal unit vector, and the equation for it uses the derivative of T(t). \( The B vector is the binormal vector, which is a crossproduct of T and N. asked 2020-10-21 For any vectors u, v and w, show that the vectors u-v, v-w and w-u form a linearly dependent set. asked 2020-10-21 Find the Euclidean distance between u and v and the cosine of the angle between those vectors. State whether that angle is acute, obtuse, or \(\displaystyle{90}^{{\circ}}$$. u = (-1, -1, 8, 0), v = (5,6,1,4) For each problem below, either prove that the mapping is linear or explain why it cannot be linear.
$$\displaystyle{1}.{f{{\left({x}_{{1}},{x}_{{2}}\right)}}}={\left({2}{x}_{{1}}-{x}_{{2}},{3}{x}_{{1}}+{x}_{{2}}\right)}$$
$$\displaystyle{2}.{L}{\left({x},{y},{z}\right)}={\left({x}+{y},{y}+{z},{z}+{5}\right)}$$
$$\displaystyle{3}.{L}{\left({x},{y}\right)}={\left({x}+{y},{0},{x}-{2}{y}\right)}$$
$$\displaystyle{4}.{f{{\left({x},{y}\right)}}}={\left({2}{x}+{y},-{3}{x}+{5}{y}\right)}$$
$$\displaystyle{5}.{f{{\left({x},{y}\right)}}}={\left({x}^{{2}},{x}+{y}\right)}$$
$$\displaystyle{6}.{L}{\left({x},{y}\right)}={\left({x},{x}+{y},-{y}\right)}$$ Determine if
$$\displaystyle{a}.{H}={\left\lbrace\frac{{{x},{y}}}{{y}}={3}{x}-{1}\right\rbrace}$$ is a subspace of R2
$$\displaystyle{b}.{H}={\left\lbrace{a}{t}+\frac{{b}}{{b}}={8}{a}\right\rbrace}$$ is a subspace of P1 Determine the area under the standard normal curve that lies between ​
(a) Upper Z equals -2.03 and Upper Z equals 2.03​,
​(b) Upper Z equals -1.56 and Upper Z equals 0​, and
​(c) Upper Z equals -1.51 and Upper Z equals 0.68. ​ ​(Round to four decimal places as​ needed.) Find the angles made by the vectors $$\displaystyle{A}={5}{i}-{2}{j}+{3}{k}$$ with the axes give a full correct answer  Let $$\displaystyle{v}_{{1}},{v}_{{2}},\ldots.,{v}_{{k}}$$ be vectors of Rn such that
$$\displaystyle{v}={c}_{{1}}{v}_{{1}}+{c}_{{2}}{v}_{{2}}+\ldots+{c}_{{k}}{v}_{{k}}={d}_{{1}}{v}_{{1}}+{d}_{{2}}{v}_{{2}}+\ldots+{d}_{{k}}{v}_{{k}}$$.
for some scalars $$\displaystyle{c}_{{1}},{c}_{{2}},\ldots.,{c}_{{k}},{d}_{{1}},{d}_{{2}},\ldots.,{d}_{{k}}$$.Prove that if $$\displaystyle{c}{i}\ne{d}{i}{f}{\quad\text{or}\quad}{s}{o}{m}{e}{i}={1},{2},\ldots.,{k}$$,
then $$\displaystyle{v}_{{1}},{v}_{{2}},\ldots.,{v}_{{k}}$$ are linearly dependent. 