Đáp án:

\(mp(P) \perp mp(Q) \Leftrightarrow \vec{n_P} = (2, -2m, 4) \perp \vec{n_Q} = (m+3, -2, 5)  \)

\( \Leftrightarrow 2(m+3) + 4m + 20 = 0 \)  
\( \Leftrightarrow 6m + 26 = 0  \Leftrightarrow m = -\frac{13}{3} \approx -4.33 \Rightarrow \boxed{B} \) 

page47

Đáp án:

\(\begin{cases} \vec{n}_D = (1, 1, 1) \\ \vec{n}_Q = (1, -1, 1) \end{cases} \implies \vec{n}_R = [\vec{n}_D, \vec{n}_Q] = (2, 0, -2).\)

pt mp(R):  \(2x - 2z + D = 0\)

\(d(O, mp(R)) = 2 \implies \frac{|D|}{\sqrt{2}} = 2 \implies D = \pm 2\sqrt{2}.\)

Vậy:  mp(R): \( \Big[ \begin{aligned} mp(R): 2x - 2z + 2\sqrt{2} = 0 \\ mp(R): 2x - 2z - 2\sqrt{2} = 0\end{aligned}.\)

Đáp án:

 mp(R):  \(2x + 4y - 2z + 2 = 0\)

page48

Đáp án:

- \( (\beta) \not\perp (\delta): \)

mpP: \( y + 2z - 4 + m(x + y - 5z - 5) = 0 \)  
mpP: \( mx + (1 + m)y + (2 - 5m)z - 4 - 5m = 0 \)

- mpP  \(\perp (\delta): \Leftrightarrow \vec{n}_P \perp \vec{n}_\delta \Leftrightarrow m + 1 + m + 2 - 5m = 0 \Leftrightarrow m = 1 \)

- mpP: \( x + 2y - 3z - 9 = 0 \Rightarrow \boxed{A} \)

page49

page50 

Đáp án:

- Mp \( P: ax + by + cz + d = 0 \)  
  Mp \( P \) qua \( M(1,0,0) \)   \( \Rightarrow a + d = 0 \Rightarrow d = -a. \)  
  Mp \( P \) qua \( N(0,0,-1) \)   \( \Rightarrow -c + d = 0 \Rightarrow c = d = -a. \)  
\( \Rightarrow \text{Mp } P: ax + by - az - a = 0 \Rightarrow \vec{n}_P = (a, b, -a). \)  

- \( \vec{n}_Q = (1, -1, 0). \)  
  Góc \(((P), (Q)) = 45^\circ \Leftrightarrow  \cos (45^\circ) = | \cos{(\vec{n_P, n_Q})}|\)

\( \Leftrightarrow \frac{|a - b|}{\sqrt{a^2 + b^2 + a^2} \sqrt{2}} = \frac{\sqrt{2}}{2} \Leftrightarrow |a - b| = \sqrt{2a^2 + b^2}. \)  

  \( \Rightarrow a^2 + b^2 - 2ab = 2a^2 + b^2 \Leftrightarrow a^2 + 2ab = 0 . \)  

- \( \left[ \begin{split} &a=0 (\text{ Chọn b = 1}): \quad &y = 0 \\ &a=-2b (\text{ Chọn b = -1}): \quad &2x - y - 2z - 2 = 0\end{split} \right. \Rightarrow \boxed{A} \)

page51