Title: MAT. Equacions d'una recta i d'un pla a l'espai
Course: Matemàtiques de 3r d'ESO

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Q: Trobeu l'equació general del pla que determinen els punts $$P=({p1},{p2},{p3})$$, $$Q=({q1},{q2},{q3})$$ i $$R=({r1},{r2},{r3})$$
- {={p2}*{q3}-{p2}*{r3}-{p3}*{q2}+{p3}*{r2}+{q2}*{r3}-{q3}*{r2}}x+{=-{p1}*{q3}+{p1}*{r3}+{p3}*{q1}-{p3}*{r1}-{q1}*{r3}+{q3}*{r1}}y+{={p1}*{q2}-{p1}*{r2}-{p2}*{q1}+{p2}*{r1}+{q1}*{r2}-{q2}*{r1}}z+{=-{p1}*{q2}*{r3}+{p1}*{q3}*{r2}+{p2}*{q1}*{r3}-{p2}*{q3}*{r1}-{p3}*{q1}*{r2}+{p3}*{q2}*{r1}}=0
- {={q2}*{r3}-{q3}*{r2}}x+{=-{q1}*{r3}+{q3}*{r1}}y+{={q1}*{r2}-{q2}*{r1}}z+{=-{p1}*{q2}*{r3}+{p1}*{q3}*{r2}+{p2}*{q1}*{r3}-{p3}*{q1}*{r2}+{p3}*{q2}*{r1}}=0
- {={q2}*{r3}-{q3}*{r2}}x+{=-{q1}*{r3}+{q3}*{r1}}y+{={q1}*{r2}-{q2}*{r1}}z+{={p1}*{q2}*{r3}-{p1}*{q3}*{r2}-{p2}*{q1}*{r3}+{p3}*{q1}*{r2}-{p3}*{q2}*{r1}}=0
- {={q2}*{r3}*2-{q3}*{r2}}x+{={q1}*{r3}-{q3}*{r1}}y+{={q1}*{r2}-2*{q2}*{r1}}z+{=-{p1}*{q2}*{r3}-{p1}*{q3}*{r2}+{p2}*{q1}*{r3}+{p3}*{q1}*{r2}-{p3}*{q2}*{r1}}=0
- {=3*{q2}*{r3}-2*{q3}*{r2}}x+{=-{q1}*{q1}+{q3}*{r1}}y+{=4*{q1}*{r2}-{q2}*{r1}}z+{={p1}*{q2}*{r3}-{p1}*{q3}*9-{p2}*{q1}*{r3}+{p3}*{q1}*{r2}+{p3}*{q2}*{r1}}=0
- {={p1}*{q3}-{p2}*{r2}-{p3}*{q2}+{p3}*{r2}+{q2}*{r3}+{q3}*{r2}}x+{={p1}*{q3}+{p1}*{r3}+{p3}*{q2}-{p3}*{r1}-{q1}*{r3}+{q3}*{r1}}y+{={p2}*{q2}-{p1}*{r2}-{p2}*{q1}+{p2}*{r1}+{q1}*{r2}-{q2}*{r1}}z+{=-{p1}*{q2}*{r3}+{q3}*{q3}*{r2}+{p2}*{q1}*{r3}-{p2}*{q3}*{r1}-{p3}*{q1}*{r2}+{p3}*{q2}*{r1}}=0
- {={p1}*{q3}-{p3}*{r3}-{p3}*{q1}+{p3}*{r2}+{q2}*{r3}-{q3}*{r2}}x+{=-{p1}*{q3}+{p2}*{r3}+{p3}*{q1}-{p2}*{r1}-{q1}*{r3}+{q3}*{r1}}y+{={p1}*{q3}-{p1}*{r2}-{p1}*{q1}+{p2}*{r1}+{q1}*{r2}-{q2}*{r1}}z+{=-{p1}*{q2}*{r2}+{p2}*{q3}*{r1}+{p2}*{q1}*{r3}-{p2}*{q3}*{r1}-{p3}*{q1}*{r2}+{p3}*{p3}*{r1}}=0
- {=5*{p2}*{q3}-3*{p2}*{r3}-{p3}*{q2}+{p3}*{r2}+{q2}*{r3}-{q3}*{r2}}x+{=-{p1}*{q2}+{p1}*{r3}+{p3}*{q1}-{p3}*{r1}-{q1}*{r3}+{q3}*{r1}}y+{={p1}*{q2}-{p1}*{r3}-{p2}*{q1}+{p2}*{r1}+{q1}*{r2}-{q2}*{r1}}z+{=-{p1}*{q2}*{r3}+{p1}*{q3}*{r2}+{p2}*{q1}*{r3}-{p2}*{q3}*{r1}-{p3}*{q1}*{r2}+{p3}*{q1}*{r1}}=0
- {={p2}+1}x+{=-{r3}+5}y+{={q2}-3}z+{=-{p3}+{q2}+{r1}}=0
A: {={p2}*{q3}-{p2}*{r3}-{p3}*{q2}+{p3}*{r2}+{q2}*{r3}-{q3}*{r2}}x+{=-{p1}*{q3}+{p1}*{r3}+{p3}*{q1}-{p3}*{r1}-{q1}*{r3}+{q3}*{r1}}y+{={p1}*{q2}-{p1}*{r2}-{p2}*{q1}+{p2}*{r1}+{q1}*{r2}-{q2}*{r1}}z+{=-{p1}*{q2}*{r3}+{p1}*{q3}*{r2}+{p2}*{q1}*{r3}-{p2}*{q3}*{r1}-{p3}*{q1}*{r2}+{p3}*{q2}*{r1}}=0 (formula; values not in the backup's text)

Q: Donat un punt $$A=(a_1,a_2,a_3)$$ i un vector $$\vec{v}=(v_1,v_2,v_3)$$ l'equació contínua de la recta és ...
- $$\frac{x-a_1}{v_1}=\frac{y-a_2}{v_2}=\frac{z-a_3}{v_3}$$
- $$\frac{x+a_1}{v_1}=\frac{y+a_2}{v_2}=\frac{z+a_3}{v_3}$$
- $$\frac{x-v_1}{a_1}=\frac{y-v_2}{a_2}=\frac{z-v_3}{a_3}$$
- $$\frac{x+v_1}{a_1}=\frac{y+v_2}{a_2}=\frac{z+v_3}{a_3}$$
A: $$\frac{x-a_1}{v_1}=\frac{y-a_2}{v_2}=\frac{z-a_3}{v_3}$$

Q: Donades les rectes $$r:\frac{x-{a1}}{{v1}}=\frac{y+{a2}}{{2}}=\frac{z-2}{{v3}}$$ i $$s:\begin{array}{ccc} {n1}x+4y-z+{n4}=0\\2x+{m2}y+z+{m4}=0\\ \end{array} $$, trobeu l'equació general d'un pla paral·lel a $$r$$ i $$s$$ que passi pel punt $$P=({p1},0,-{p3})$$
- {={m2}*{n1}+{n1}*{v3}+{v3}-16}x+{={m2}*{n1}*{v1}+{m2}*{v3}+8*{v1}+4*{v3}}y+{=-{m2}-{n1}*{v1}-2*{v1}-8}z+{=-2*{m2}*{n1}*{p1}-2*{m2}*{p3}-{n1}*{p1}*{v3}-{n1}*{p3}*{v1}-2*{p1}*{v3}+16*{p1}-2*{p3}*{v1}-{p3}}=0
- {=2*{m2}*{n1}+{n1}*{v3}+2*{v3}-16}x+{=-{m2}*{n1}*{v1}+{m2}*{v3}+8*{v1}+4*{v3}}y+{=-2*{m2}-{n1}*{v1}-2*{v1}-8}z+{=-2*{m2}*{n1}*{p1}-2*{m2}*{p3}-{n1}*{p1}*{v3}-{n1}*{p3}*{v1}-2*{p1}*{v3}+16*{p1}-2*{p3}*{v1}-8*{p3}}=0
- {={v1}}x+2y+{={v3}}z+{={v3}*{p3}-{v1}*{p1}}=0
- {={v1}}x+2y+{={v3}}z+{=-{v3}*{p3}+{v1}*{p1}}=0
- {=-{v1}}x+2y+{=-{v3}}z+{={v3}*{p3}-{v1}*{p1}+5}=0
- {={v1}}x+2y+{={v3}}z+{={v3}*{p3}-{v1}*{p1}-2}=0
- {=2*{m2}+{n1}+2*{v3}-23}x+{=-{m2}*{n1}*9+{m2}*{v3}+8*{v1}+4*{v3}+12}y+{=-2*{m2}-{n1}*{v1}+15}z+{={n1}*{p1}-2*{m2}*{p3}-{n1}*{p1}*{v3}-{n1}*{p3}-2*{p1}*{v3}+16*{p1}-2*{p3}*{v1}-8*{p3}}=0
- {=2*{v1}}x+9y+{={v3}+4}z+{=5*{v3}*{p3}-3*{v1}*{p1}}=0
- No existeix tal pla
- {=3+2*{m2}*{n1}+{n1}*{v3}+2*{v3}-16}x+{=5-{m2}*{n1}*{v1}+{m2}*{v3}+8*{v1}+4*{v3}}y+{=-4-2*{m2}-{n1}*{v1}-2*{v1}-8}z+{=-2*{m2}*{n1}*{p1}-2*{m2}*{p3}-{n1}*{p1}*{v3}-{n1}*{p3}*{v1}-2*{p1}*{v3}+16*{p1}-2*{p3}*{v1}-8*{p3}+12}=0
- {=7+2*{m2}*{n1}+{n1}*{v3}+2*{v3}-16}x+{=-4-{m2}*{n1}*{v1}+{m2}*{v3}+8*{v1}+4*{v3}}y+{=-6-2*{m2}-{n1}*{v1}-2*{v1}-8}z+{=-2*{m2}*{n1}*{p1}-2*{m2}*{p3}-{n1}*{p1}*{v3}-{n1}*{p3}*{v1}-2*{p1}*{v3}+16*{p1}-2*{p3}*{v1}-8*{p3}-3}=0
A: {=2*{m2}*{n1}+{n1}*{v3}+2*{v3}-16}x+{=-{m2}*{n1}*{v1}+{m2}*{v3}+8*{v1}+4*{v3}}y+{=-2*{m2}-{n1}*{v1}-2*{v1}-8}z+{=-2*{m2}*{n1}*{p1}-2*{m2}*{p3}-{n1}*{p1}*{v3}-{n1}*{p3}*{v1}-2*{p1}*{v3}+16*{p1}-2*{p3}*{v1}-8*{p3}}=0 (formula; values not in the backup's text)

Q: Donat un punt $$A=(a_1,a_2,a_3)$$ i dos vectors directors $$\vec{v}=(v_1,v_2,v_3)$$ i $$\vec{w}=(w_1,w_2,w_3)$$ les equacions paramètriques del pla són ...
- $$ \left| \begin{array}{ccc} x+a_1 & y+a_2 & z+a_3\\v_1 & v_2 & v_3\\w_1 & w_2 & w_3\\ \end{array} \right|=0$$
- $$\frac{x+a_1}{v_1}=\frac{y+a_2}{v_2}=\frac{z+a_3}{v_3}$$
- $$\frac{x-a_1}{v_1}=\frac{y-a_2}{v_2}=\frac{z-a_3}{v_3}$$
- $$ \left| \begin{array}{ccc} x-a_1 & y-a_2 & z-a_3\\v_1 & v_2 & v_3\\w_1 & w_2 & w_3\\ \end{array} \right| =0$$
- $$ \left\{ \begin{array}{ccc} x=a_1+ \lambda v_1+ \mu w_1\\y=a_2+ \lambda v_2+ \mu w_2\\z=a_3+ \lambda v_3+ \mu w_3\\ \end{array} \right\}$$
- $$ \left\{ \begin{array}{ccc} x=a_1+ \lambda v_1\\y=a_2+ \lambda v_2\\z=a_3+ \lambda v_3\\ \end{array} \right\}$$
A: $$ \left\{ \begin{array}{ccc} x=a_1+ \lambda v_1+ \mu w_1\\y=a_2+ \lambda v_2+ \mu w_2\\z=a_3+ \lambda v_3+ \mu w_3\\ \end{array} \right\}$$

Q: L'equació general (o implícita, o cartesiana) d'una recta a l'espai és de la forma ...
- $$Ax+By+C=0$$
- $$Ax+By+Cz+D=0$$
- $$ \left\{\begin{array}{ccc} A_1x+B_1y+C_1=0\\A_2x+B_2y+C_2=0\\ \end{array} \right\} $$
- $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\\ \end{array} \right\}$$
- $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1=0\\A_2x+B_2y+C_2=0\\A_3x+B_3y+C_3=0\\ \end{array} \right\}$$
- $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\\A_3x+B_3y+C_3z+D_3=0 \end{array} \right\}$$
- $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1=0\\A_2x+B_2y+C_2=0\\A_3x+B_3y+C_3=0\\A_4x+B_4y+C_4=0 \end{array} \right\}$$
- $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\\A_3x+B_3y+C_3z+D_3=0\\A_4x+B_4y+C_4z+D_4=0 \end{array} \right\}$$
A: $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\\ \end{array} \right\}$$

Q: L'equació general (o implícita, o cartesiana) d'un pla a l'espai és de la forma ...
- $$Ax+By+C=0$$
- $$Ax+By+Cz+D=0$$
- $$ \left\{\begin{array}{ccc} A_1x+B_1y+C_1=0\\A_2x+B_2y+C_2=0\\ \end{array} \right\}$$
- $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\\ \end{array} \right\}$$
- $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1=0\\A_2x+B_2y+C_2=0\\A_3x+B_3y+C_3=0\\ \end{array} \right\}$$
- $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\\A_3x+B_3y+C_3z+D_3=0 \end{array} \right\}$$
- $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1=0\\A_2x+B_2y+C_2=0\\A_3x+B_3y+C_3=0\\A_4x+B_4y+C_4=0 \end{array} \right\}$$
- $$\left\{ \begin{array}{ccc} A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\\A_3x+B_3y+C_3z+D_3=0\\A_4x+B_4y+C_4z+D_4=0 \end{array} \right\}$$
A: $$Ax+By+Cz+D=0$$

Q: Donat un punt $$A=(a_1,a_2,a_3)$$ i dos vectors directors $$\vec{v}=(v_1,v_2,v_3)$$ i $$\vec{w}=(w_1,w_2,w_3)$$ l'equació general del pla es pot trobar a partir de ...
- $$ \left| \begin{array}{ccc} x+a_1 & y+a_2 & z+a_3\\v_1 & v_2 & v_3\\w_1 & w_2 & w_3\\ \end{array} \right|=0$$
- $$\frac{x+a_1}{v_1}=\frac{y+a_2}{v_2}=\frac{z+a_3}{v_3}$$
- $$\frac{x-a_1}{v_1}=\frac{y-a_2}{v_2}=\frac{z-a_3}{v_3}$$
- $$ \left| \begin{array}{ccc} x-a_1 & y-a_2 & z-a_3\\v_1 & v_2 & v_3\\w_1 & w_2 & w_3\\ \end{array} \right| =0$$
- $$ \left\{ \begin{array}{ccc} x=a_1+ \lambda v_1+ \mu w_1\\y=a_2+ \lambda v_2+ \mu w_2\\z=a_3+ \lambda v_3+ \mu w_3\\ \end{array} \right\}$$
- $$ \left\{ \begin{array}{ccc} x=a_1+ \lambda v_1\\y=a_2+ \lambda v_2\\z=a_3+ \lambda v_3\\ \end{array} \right\}$$
A: $$ \left| \begin{array}{ccc} x-a_1 & y-a_2 & z-a_3\\v_1 & v_2 & v_3\\w_1 & w_2 & w_3\\ \end{array} \right| =0$$

Q: Donat el pla $${A}x+{B}y-{C}z-{D}=0$$ calculeu l'equació de la recta perpendicular que passa pel punt $$P=(-{p1},{p2},-{p3})$$

A. $$\frac{x+{{p1}}}{{A}}=\frac{y-{{p2}}}{{B}}=\frac{z+{{p3}}}{-{C}}$$

B. $$\frac{x-{{A}}}{-{p1}}=\frac{y-{{B}}}{{p2}}=\frac{z+{{C}}}{-{p3}}$$

C. $$\frac{x+{{A}}}{-{p1}}=\frac{y+{{B}}}{{p2}}=\frac{z-{{C}}}{-{p3}}$$

D. $$\frac{x-{{p1}}}{{A}}=\frac{y+{{p2}}}{{B}}=\frac{z-{{p3}}}{-{C}}$$

E. No existeix tal recta

F. $$-{p1}x+{p2}y-{p3}z-{={D}+{A}}=0$$

G. $${p1}x-{p2}y+{p3}z+{={D}+{A}}=0$$

H. $$-{A}x-{B}y+{C}z+{D}=0$$

0. 0
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- {={A}-{A}}
A: A

Q: Calculeu l'equació de la recta que passa per la intersecció del pla $$\pi: {A}x-{B}y+{C}z+{=(-3*{A}+2*{B}+4*{C})*{t}}=0$$ amb la recta $$r: \frac{x-{=9*{t}}}{-6}=\frac{y}{2}=\frac{z+{=9*{t}}}{5}$$ i és paral·lela a la recta $$s:\left\{ \begin{array}{ccc} {n1}x+y-{n3}z+{n4}=0\\3x+{m2}y+{m4}=0\\ \end{array}\right\}$$

A. $$\frac{x-{=3*{t}}}{{={m2}*{n3}}}=\frac{y-{=2*{t}}}{{=-3*{n3}}}=\frac{z+{=4*{t}}}{{={m2}*{n1}-3}}$$

B. $$\frac{x-{=3*{t}}}{-6}=\frac{y-{=2*{t}}}{2}=\frac{z+{=4*{t}}}{5}$$

C. $$\frac{x-{=3*{t}}}{{=-{m2}*{n3}}}=\frac{y-{=2*{t}}}{{=-3*{n3}}}=\frac{z+{=4*{t}}}{{={m2}*{n1}-3}}$$

D. $$\frac{x-{=3*{t}}}{{={m2}*{n3}}}=\frac{y-{=2*{t}}}{{=3*{n3}}}=\frac{z+{=4*{t}}}{{={m2}*{n1}-3}}$$

E. $$\frac{x-{=3*{t}}}{{={m2}*{n3}}}=\frac{y-{=2*{t}}}{{=-3*{n3}}}=\frac{z+{=4*{t}}}{{=-{m2}*{n1}+3}}$$

F. $$\frac{x-{=3*{t}}}{{=1+{m2}*{n3}}}=\frac{y-{=2*{t}}}{{=-3*{n3}+2}}=\frac{z+{=4*{t}}}{{={m2}*{n1}-1}}$$

G. $$\frac{x-{=3*{t}}}{{=-1-{m2}*{n3}}}=\frac{y-{=2*{t}}}{{=-3*{n3}+2}}=\frac{z+{=4*{t}}}{{={m2}*{n1}-1}}$$

H. $$\frac{x-{=3*{t}}}{{=1+{m2}*{n3}}}=\frac{y-{=2*{t}}}{{=3*{n3}-2}}=\frac{z+{=4*{t}}}{{={m2}*{n1}-1}}$$ç

I. $$\frac{x-{=3*{t}}}{{=1+{m2}*{n3}}}=\frac{y-{=2*{t}}}{{=-3*{n3}+2}}=\frac{z+{=4*{t}}}{{=-{m2}*{n1}+1}}$$

J. No existeix tal recta

K. $$\frac{x+{=3*{t}}}{{={m2}*{n3}}}=\frac{y+{=2*{t}}}{{=-3*{n3}}}=\frac{z-{=4*{t}}}{{={m2}*{n1}-3}}$$

L. $$\frac{x+{=3*{t}}}{-6}=\frac{y+{=2*{t}}}{2}=\frac{z-{=4*{t}}}{5}$$

M. $$\frac{x+{=3*{t}}}{{=-{m2}*{n3}}}=\frac{y+{=2*{t}}}{{=-3*{n3}}}=\frac{z-{=4*{t}}}{{={m2}*{n1}-3}}$$

N. $$\frac{x+{=3*{t}}}{{={m2}*{n3}}}=\frac{y+{=2*{t}}}{{=3*{n3}}}=\frac{z-{=4*{t}}}{{={m2}*{n1}-3}}$$

O. $$\frac{x+{=3*{t}}}{{={m2}*{n3}}}=\frac{y+{=2*{t}}}{{=-3*{n3}}}=\frac{z-{=4*{t}}}{{=-{m2}*{n1}+3}}$$

P. $$\frac{x+{=3*{t}}}{{=1+{m2}*{n3}}}=\frac{y+{=2*{t}}}{{=-3*{n3}+2}}=\frac{z-{=4*{t}}}{{={m2}*{n1}-1}}$$

Q. $$\frac{x+{=3*{t}}}{{=-1-{m2}*{n3}}}=\frac{y+{=2*{t}}}{{=-3*{n3}+2}}=\frac{z-{=4*{t}}}{{={m2}*{n1}-1}}$$

R. $$\frac{x+{=3*{t}}}{{=1+{m2}*{n3}}}=\frac{y+{=2*{t}}}{{=3*{n3}-2}}=\frac{z-{=4*{t}}}{{={m2}*{n1}-1}}$$

S. $$\frac{x+{=3*{t}}}{{=1+{m2}*{n3}}}=\frac{y+{=2*{t}}}{{=-3*{n3}+2}}=\frac{z-{=4*{t}}}{{=-{m2}*{n1}+1}}$$
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A: A

Q: Troba un pla paral·lel a $${A}x-{B}y-{C}z+{D}=0$$ que passi pel punt $$P=({p1},{p2},{p3})$$
- {A}x-{B}y-{C}z+{={D}+{A}*{p1}+{B}*{p2}+{C}*{p3}}=0
- {A}x-{B}y-{C}z+{=-{D}-{A}*{p1}-{B}*{p2}-{C}*{p3}}=0
- {A}x-{B}y-{C}z+{={A}*{p1}+{B}*{p2}+{C}*{p3}}=0
- {A}x-{B}y-{C}z+{=-{A}*{p1}+{B}*{p2}+{C}*{p3}}=0
- {A}x-{B}y-{C}z+{={D}-{A}*{p1}-{B}*{p2}-{C}*{p3}}=0
- {A}x-{B}y-{C}z+{=-{D}+{A}*{p1}+{B}*{p2}+{C}*{p3}}=0
- -{A}x+{B}y+{C}z+{=-{A}*{p1}-{B}*{p2}-{C}*{p3}}=0
- -{A}x+{B}y+{C}z+{=-2*{A}*{p1}-{B}*{p2}-{C}*{p3}}=0
- -{A}x+{B}y+{C}z+{=-{A}*{p1}+{B}*{p2}-{C}*{p3}}=0
- -{A}x+{B}y+{C}z+{=-{A}*{p1}-{B}*{p2}+2*{C}*{p3}}=0
- -{A}x+{B}y+{C}z+{=-2*{A}*{p1}+3*{B}*{p2}-{C}*{p3}}=0
A: {A}x-{B}y-{C}z+{=-{A}*{p1}+{B}*{p2}+{C}*{p3}}=0 (formula; values not in the backup's text)

Q: Calculeu $$k$$ i $$m$$ perquè la recta $$\left\{ \begin{array}{ccc} x={p1}+{v1}\lambda\\y={p2}+m\lambda\\z={p3}+{={s}*{v1}}\lambda\\ \end{array}\right\} $$ i el pla $${A}x+{={t}*{A}}y+kz+{D}=0$$ siguin perpendiculars. Anoteu com a resposta el valor $$1000k+m$$
A: 1000*{s}*{A}+{t}*{v1} (formula; values not in the backup's text)

Q: Seleccioneu els plans que són paral·lels al pla $${={a}*2}x-{={b}*2}y-{={c}*2}z+{={d}*2}=0$$
- {={a}}x-{={b}}y-{={c}*2}z-{={d}*2}=0
- {={a}*4}x+{={b}*4}y+{={c}*4}z+{={d}}=0
- -{={a}*5}x+{={b}*5}y+{={c}*5}z+{={d}*5}=0
- -{={a}*3}x+{={b}*3}y+{={c}*3}z-{={d}*3}=0
- -{={a}*3}x+{={b}*3}y+{={c}*3}z+{={d}*4}=0
- {={a}}x-{={b}}y-{={c}}z+{={d}}=0
- {={a}*3}x-{={b}*3}y-{={c}*3}z+{={d}+1}=0
- {={a}*2}x-{={b}*2}y-{={c}*2}z+{={d}*2-10}=0
A: -{={a}*5}x+{={b}*5}y+{={c}*5}z+{={d}*5}=0 (formula; values not in the backup's text) / -{={a}*3}x+{={b}*3}y+{={c}*3}z+{={d}*4}=0 (formula; values not in the backup's text) / {={a}*3}x-{={b}*3}y-{={c}*3}z+{={d}+1}=0 (formula; values not in the backup's text) / {={a}*2}x-{={b}*2}y-{={c}*2}z+{={d}*2-10}=0 (formula; values not in the backup's text)

Q: Donada la recta $${ 2x-y+z-3=0 \brace x+3y-5z+8=0}$$
Marqueu els resultats correctes
- (2,-1,1) és un vector director de r
- (1,3,-5) és un vector director de r
- (2,11,7) és un vector director de r
- (-4,-22,-14) és un vector director de r
- (1,3,-5) és un punt de r
- (2,-1,1) és un punt de r
- (2,-1,1) és un punt de r
- (1,2,3) és un punt de r
A: (2,11,7) és un vector director de r; (-4,-22,-14) és un vector director de r; (1,2,3) és un punt de r
If partly right: 1

Q: Assenyaleu els punts que pertanyen a la recta $$r:
\frac{x-5}{2}=y+3=\frac{z-2}{4}$$
- $$(2,0,4)$$
- $$(2,1,4)$$
- $$(-5,3,-2)$$
- $$(5,-3,2)$$
- $$(1,-5,-6)$$
A: $$(5,-3,2)$$; $$(1,-5,-6)$$
If partly right: 1

Q: Trobeu l'equació de la recta paral·lela a $$\left\{ \begin{array}{ccc} {A1}x+{B1}y-z+{D1}=0\\{A2}x-{C2}z+{D2}=0\\ \end{array} \right\}$$ que passi pel punt $$P=(-{p1},{p2},0)$$

A. $$\frac{x-{{p1}}}{{{={B1}*{C2}}}}=\frac{y+{{p2}}}{{{=-{A1}*{C2}-{A2}}}}=\frac{z{{}}}{{{={A2}*{B1}}}}$$

B. $$\frac{x+{{p1}}}{{{=-{B1}*{C2}}}}=\frac{y-{{p2}}}{{{={A1}*{C2}-{A2}}}}=\frac{z{{}}}{{{=-{A2}*{B1}}}}$$

C. No existeix tal recta

D. $$\frac{x+{{p1}}}{{{=5-{B1}*{C2}}}}=\frac{y-{{p2}}}{{{={A1}*{C2}-{A2}+5}}}=\frac{z{{}}}{{{=2-{A2}*{B1}}}}$$

E. $$\frac{x-{{p1}}}{{{=5-{B1}*{C2}}}}=\frac{y+{{p2}}}{{{={A1}*{C2}-{A2}+5}}}=\frac{z{{}}}{{{=2-{A2}*{B1}}}}$$

F. $$\frac{x+{{p1}}}{{A1}}=\frac{y-{{p2}}}{{B1}}=\frac{z{{}}}{{-1}}$$

G. $$\frac{x-{{p1}}}{{A1}}=\frac{y+{{p2}}}{{B1}}=\frac{z{{}}}{{-1}}$$

H. $$\frac{x+{{p1}}}{{{=-{B1}*{C2}}}}=\frac{y-{{p2}}}{{{={A1}*{C2}-4}}}=\frac{z{{}}}{{{={A2}-3}}}$$

I. $$\frac{x-{{p1}}}{{{=-{B1}*{C2}}}}=\frac{y+{{p2}}}{{{={A1}*{C2}-4}}}=\frac{z{{}}}{{{={A2}-3}}}$$

J. $${{A1}x+{B1}y-z-{p1}=0   \brace   {A2}x-{C2}z+{p2}=0}$$
- A
- B
- C
- D
- E
- F
- G
- H
- {={A1}/{A1}}
- J
A: B

Q: Trobeu un pla perpendicular al pla $${A1}x-4y+{C1}z+{D1}=0$$ que contingui la recta $$\left\{\begin{array}{ccc} 3x+{B2}y-{C2}z+{={C2}*3}=0\\-{A3}x+y-{=2*{C2}}=0 \end{array}\right\} $$
- {=3*{A3}*{B2}-{A3}*{C1}-12}x+{={A1}*{A3}*{B2}-3*{A1}+{B2}}y+{={B2}*{A3}+4}z+{=-3*{A1}*{A3}*{B2}*{C2}+5*{A1}*{A3}*{B2}-3*{A1}*{A3}+6*{A1}*{C2}-8*{B2}-2*{C1}*{C2}}=0
- No existeix tal pla
- {=9*{A3}*{B2}-7*{A3}*{C1}-11}x+{=5*{A1}*{A3}*{B2}-4*{A1}+{C1}}y+{=3*{A1}*{A3}-3}z+{=-3*{A1}*{A3}*{B2}*{C2}-{A1}*{A3}*{B2}-3*{A1}*{A3}+6*{A1}*{C2}-9*{B2}-2*{C1}*{C2}+3}=0
- {=4*{A3}*{B2}-{A3}*{C1}-9}x+{={A1}*{A3}*{B2}-4*{A1}+{C1}}y+{={A1}*{A3}+7}z+{=-2*{A1}*{A3}*{B2}*{C2}-2*{A1}*{A3}*{B2}-3*{A1}*{A3}+6*{A1}*{C2}-8*{B2}-2*{C1}*{C2}}=0
- {=4*{A3}*{B2}+{A3}*{C1}*{C2}+12}x+{={A1}*{A3}*{B2}+3*{A1}-{C1}*{C2}}y+{=-{A1}*{A3}*{C2}-4*{C2}}z+{=3*{A1}*{A3}*{C2}-6*{A1}*{C2}+8*{B2}*{C2}+2*{C1}*{C2}*{C2}+12*{C2}}=0
- {={A3}*{B2}+{A3}*{C1}-18}x+{={A1}*{A3}*{B2}+3*{A1}+{C1}-4}y+{=-{A1}*{A3}+23}z+{=-10*{A1}*{A3}*{B2}*{C2}-9*{A1}*{A3}*{B2}+15*{A1}*{A3}+6*{A1}*{C2}-8*{B2}-2*{C1}*{C2}}=0
- {=5*{A3}*{B2}-6*{A3}*{C1}-7}x+{=3*{A1}*{A3}*{B2}-4*{A1}+2*{C1}}y+{=-2*{A1}*{A3}+7}z+{=-{A1}*{A3}*{B2}*{C2}-3*{A1}*{A3}*{B2}-5*{A1}*{A3}+8*{A1}*{C2}-9*{B2}-2*{C1}*{C2}}=0
- {=4*{A3}*{B2}-{A3}*{C1}-12}x+{={A1}*{A3}*{B2}-3*{A1}+{C1}}y+{={A1}*{A3}+4}z+{=-2*{A1}*{A3}*{B2}*{C2}-2*{A1}*{A3}*{B2}-3*{A1}*{A3}+6*{A1}*{C2}-8*{B2}-2*{C1}*{C2}}=0
A: {=4*{A3}*{B2}+{A3}*{C1}*{C2}+12}x+{={A1}*{A3}*{B2}+3*{A1}-{C1}*{C2}}y+{=-{A1}*{A3}*{C2}-4*{C2}}z+{=3*{A1}*{A3}*{C2}-6*{A1}*{C2}+8*{B2}*{C2}+2*{C1}*{C2}*{C2}+12*{C2}}=0 (formula; values not in the backup's text)