Problema 3
\(\vec{A}=4i-3j+k\)\(=>4,\ Ay=-3\)\(\ Az=1\)
\(\vec{V}=-i+1j+4k\)\(=>Bx=-1,By=1\)
\(\vec{A.\vec{B}}=\left(4\right)\left(-1\right)+\left(-3\right)\left(1\right)+\left(1\right)\left(4\right)\)
\(=-4-3+4=-3\)
\(\cos\theta=\frac{\vec{A.\vec{B}}}{\left|\vec{A}\right|\left|\vec{B}\right|}=\frac{-3}{\left(\sqrt{26}\right)\left(\sqrt{18}\right)}=\frac{-3}{\sqrt{468}}=\cos\theta^{-1_{_{ }}}\left(\frac{-3}{\sqrt{468}}\right)=\theta=97.97^{ }\)
\(\left|\vec{A}\right|=\sqrt{Ax+Ay^{2_{ }}+Az^2}\) \(\left|\vec{A}\right|\sqrt{\left(4\right)^2+\left(-3\right)^2+\left(1\right)^2}\) \(=\sqrt{16+9+1}\) \(=\sqrt{26}\) \(=5.09\)
\(\left|\vec{B}\right|=\sqrt{Bx^2+By^2+Bz^2}\) \(=\left|\vec{B}\right|\sqrt{\left(-1\right)^2+\left(1\right)^2+\left(4\right)^2}\) \(=\sqrt{1+1+16}\) \(\sqrt{18}\) \(=4.24\)
\(\vec{Ax\vec{B}}=i\left[\left(-3\right)\left(4\right)-\left(1\right)\left(1\right)\right]-j\left[\left(4\right)\left(4\right)-\left(1\right)\left(-1\right)\right]+k\left[\left(4\right)\left(1\right)-\left(-3\right)\left(-1\right)\right]=\)
\(i\left[12-1\right]-j\left[16+1\right]+k\left[4-3\right]=\)
\(11i-17j+k\)