[ u ⃗ v ⃗ w ⃗ ] 2 = ∣ u ⃗ ⋅ u ⃗ u ⃗ ⋅ v ⃗ u ⃗ ⋅ w ⃗ v ⃗ ⋅ u ⃗ v ⃗ ⋅ v ⃗ v ⃗ ⋅ w ⃗ w ⃗ ⋅ u ⃗ w ⃗ ⋅ v ⃗ w ⃗ ⋅ w ⃗ ∣ = ∣ 1 u ⃗ ⋅ v ⃗ 1 v ⃗ ⋅ u ⃗ 1 1 1 1 4 ∣ = 2 [\vec{u} \; \vec{v} \; \vec{w}]^{2}
= \begin{vmatrix}
\vec{u} \cdot \vec{u} & \vec{u} \cdot \vec{v} & \vec{u} \cdot \vec{w} \\
\vec{v} \cdot \vec{u} & \vec{v} \cdot \vec{v} & \vec{v} \cdot \vec{w} \\
\vec{w} \cdot \vec{u} & \vec{w} \cdot \vec{v} & \vec{w} \cdot \vec{w}
\end{vmatrix}
= \begin{vmatrix}
1 & \vec{u} \cdot \vec{v} & 1 \\
\vec{v} \cdot \vec{u} & 1 & 1 \\
1 & 1 & 4
\end{vmatrix} = 2 [ u v w ] 2 = u ⋅ u v ⋅ u w ⋅ u u ⋅ v v ⋅ v w ⋅ v u ⋅ w v ⋅ w w ⋅ w = 1 v ⋅ u 1 u ⋅ v 1 1 1 1 4 = 2 = 1 ( 3 ) − u ⃗ ⋅ v ⃗ ( 4 u ⃗ ⋅ v ⃗ − 1 ) + 1 ( u ⃗ ⋅ v ⃗ − 1 ) = 2 =1(3)-\vec{u} \cdot \vec{v}(4 \vec{u} \cdot \vec{v}-1)+1(\vec{u} \cdot \vec{v}-1)=2 = 1 ( 3 ) − u ⋅ v ( 4 u ⋅ v − 1 ) + 1 ( u ⋅ v − 1 ) = 2
= 3 − 4 ( u ⃗ ⋅ v → ) 2 + u → ⋅ v → + u → ⋅ v → − 1 = 2 = − 4 ( u → ⋅ v → ) 2 + 2 u → ⋅ v → = 0 =3-4(\vec{u} \cdot \overrightarrow{\mathrm{v}})^{2}+\overrightarrow{\mathrm{u}} \cdot \overrightarrow{\mathrm{v}}+\overrightarrow{\mathrm{u}} \cdot \overrightarrow{\mathrm{v}}-1=2=-4(\overrightarrow{\mathrm{u}} \cdot \overrightarrow{\mathrm{v}})^{2}+2 \overrightarrow{\mathrm{u}} \cdot \overrightarrow{\mathrm{v}}=0 = 3 − 4 ( u ⋅ v ) 2 + u ⋅ v + u ⋅ v − 1 = 2 = − 4 ( u ⋅ v ) 2 + 2 u ⋅ v = 0
( u ⃗ ⋅ v ⃗ ) ( 2 − 4 ( u ⃗ ⋅ v ⃗ ) ) = 0 ; u ⃗ ⋅ v ⃗ = 0 , u ⃗ ⋅ v ⃗ = 1 2 (\vec{u} \cdot \vec{v})(2-4(\vec{u} \cdot \vec{v}))=0 ; \vec{u} \cdot \vec{v}=0, \vec{u} \cdot \vec{v}=\frac{1}{2} ( u ⋅ v ) ( 2 − 4 ( u ⋅ v )) = 0 ; u ⋅ v = 0 , u ⋅ v = 2 1
∣ 3 u ⃗ + 5 v ⃗ ∣ 2 = 9 ∣ u ⃗ ∣ 2 + 30 u ⃗ ⋅ v ⃗ + 25 ∣ v ⃗ ∣ 2 = 9 + 15 + 25 = 49 |3 \vec{u}+5 \vec{v}|^{2}=9|\vec{u}|^{2}+30 \vec{u} \cdot \vec{v}+25|\vec{v}|^{2}=9+15+25=49 ∣3 u + 5 v ∣ 2 = 9∣ u ∣ 2 + 30 u ⋅ v + 25∣ v ∣ 2 = 9 + 15 + 25 = 49
∴ ∣ 3 u ⃗ + 5 v ⃗ ∣ = 7 \therefore|3 \vec{u}+5 \vec{v}|=7 ∴ ∣3 u + 5 v ∣ = 7