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Note that $\\boldsymbol{a}\\times \\boldsymbol{b}$ is a vector which is perpendicular to both $\\boldsymbol{a}$ and $\\boldsymbol{b}$ and hence to the plane through the origin containing $\\boldsymbol{a}$ and $\\boldsymbol{b}$.

So if $\\boldsymbol{c}$ is perpendicular to $\\boldsymbol{a} \\times \\boldsymbol{b}$, i.e. $(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} = 0$, it must lie on the same plane.

Now

\\begin{align}
\\boldsymbol{a} \\times \\boldsymbol{b} &= \\var{vector(x1,x2,x3)} \\times \\var{vector(y1,y2,y3)} \\\\[1em]
&= \\simplify[]{vector({x2}*{y3}-{x3}*{y2}, {x3}*{y1}-{x1}*{y3}, {x1}*{y2}-{x2}*{y1})} \\\\[1em]
&= \\var{vector(w1,w2,w3)}
\\end{align}

Hence

\\begin{align}
(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} &= \\var{vector(w1,w2,w3)} \\boldsymbol{\\cdot} \\begin{pmatrix} \\var{z1} \\\\ \\var{z2} \\\\ \\lambda \\end{pmatrix} \\\\[1em]
&= \\simplify[]{{w1}*{z1}+{w2}*{z2}+{w3}*lambda} \\\\[1em]
&= \\simplify{{w1*z1+w2*z2}+{w3}*lambda}
\\end{align}

We now require a value of $\\lambda$ so that $(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c}=0$.

\\begin{align}
&&(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} &= 0 \\\\
\\implies &&\\simplify{{w1*z1+w2*z2}+{w3}*lambda} &= 0 \\\\
\\implies &&\\lambda &= \\simplify[std]{{-w1*z1-w2*z2}/{w3}}
\\end{align}

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You are given three vectors$,

\\begin{align}
\\boldsymbol{a} &= \\var{vector(x1,x2,x3)}, &
\\boldsymbol{b} &= \\var{vector(y1,y2,y3)}, &
\\boldsymbol{c} &= \\begin{pmatrix} \\var{z1} \\\\ \\var{z2} \\\\ \\lambda \\end{pmatrix}
\\end{align}

where $\\lambda$ is a parameter to be determined.

Find the value of $\\lambda$ such that $\\boldsymbol{a}$, $\\boldsymbol{b}$ and $\\boldsymbol{c}$ all lie on the same plane through the origin.

$\\lambda=$ [[0]].

Enter your answer as a fraction or integer and not a decimal.

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