( a ⃗ + b ⃗ ) ∙ ( a ⃗ − b ⃗ ) = ∣ a ⃗ ∣ 2 − ∣ b ⃗ ∣ 2 = 0 \left( {\vec{a}} + {\vec{b}} \right) \bullet \left( {\vec{a}} - {\vec{b}} \right) = \left| {\vec{a}} \right| ^{2} - \left| {\vec{b}} \right| ^{2} = 0 ( a + b ) ∙ ( a − b ) = ∣ a ∣ 2 − b 2 = 0 ∣ a ⃗ ∣ = ∣ b ⃗ ∣ \left| \vec{a} \right| = \left| \vec{b} \right| ∣ a ∣ = b ⋯ \cdots ⋯ ⋯ \cdots ⋯ ∣ a ⃗ + b ⃗ ∣ = 6 \left| {\vec{a}} + {\vec{b}} \right| = 6 a + b = 6 ∣ a ⃗ + b ⃗ ∣ 2 \left| {\vec{a}} + {\vec{b}} \right| ^{2} a + b 2 = ( a ⃗ + b ⃗ ) ∙ ( a ⃗ + b ⃗ ) = \left( {\vec{a}} + {\vec{b}} \right) \bullet \left( {\vec{a}} + {\vec{b}} \right) = ( a + b ) ∙ ( a + b ) = ∣ a ⃗ ∣ 2 + ∣ b ⃗ ∣ 2 + 2 ( a ⃗ ∙ b ⃗ ) = \left| \vec{a} \right| ^{2} + \left| \vec{b} \right| ^{2} + 2 \left( \vec{a} \bullet \vec{b} \right) = ∣ a ∣ 2 + b 2 + 2 ( a ∙ b ) ∣ a ⃗ ∣ 2 + ∣ b ⃗ ∣ 2 + 2 ( a ⃗ ∙ b ⃗ ) = 36 \left| \vec{a} \right| ^{2} + \left| \vec{b} \right| ^{2} + 2 \left( \vec{a} \bullet \vec{b} \right) = 36 ∣ a ∣ 2 + b 2 + 2 ( a ∙ b ) = 36 ∣ a ⃗ ∣ 2 + ∣ a ⃗ ∣ 2 + 2 ( a ⃗ ∙ b ⃗ ) = 36 \left| \vec{a} \right| ^{2} + \left| \vec{a} \right| ^{2} + 2 \left( \vec{a} \bullet \vec{b} \right) = 36 ∣ a ∣ 2 + ∣ a ∣ 2 + 2 ( a ∙ b ) = 36 ∵ \because ∵ ∴ \therefore ∴ ∣ a ⃗ ∣ 2 + a ⃗ ∙ b ⃗ = 18 \left| \vec{a} \right| ^{2} + \vec{a} \bullet \vec{b} = 18 ∣ a ∣ 2 + a ∙ b = 18 ⋯ \cdots ⋯ ⋯ \cdots ⋯ ∣ 2 a ⃗ − b ⃗ ∣ = 9 \left| 2 {\vec{a}} - {\vec{b}} \right| = 9 2 a − b = 9 ∣ 2 a ⃗ − b ⃗ ∣ 2 \left| 2 {\vec{a}} - {\vec{b}} \right| ^{2} 2 a − b 2 = ( 2 a ⃗ − b ⃗ ) ∙ ( 2 a ⃗ − b ⃗ ) = \left( 2 {\vec{a}} - {\vec{b}} \right) \bullet \left( 2 {\vec{a}} - {\vec{b}} \right) = ( 2 a − b ) ∙ ( 2 a − b ) = 4 ∣ a ⃗ ∣ 2 + ∣ b ⃗ ∣ 2 − 4 ( a ⃗ ∙ b ⃗ ) = 4 \left| \vec{a} \right| ^{2} + \left| \vec{b} \right| ^{2} - 4 \left( \vec{a} \bullet \vec{b} \right) = 4 ∣ a ∣ 2 + b 2 − 4 ( a ∙ b ) 4 ∣ a ⃗ ∣ 2 + ∣ b ⃗ ∣ 2 − 4 ( a ⃗ ∙ b ⃗ ) = 81 4 \left| \vec{a} \right| ^{2} + \left| \vec{b} \right| ^{2} - 4 \left( \vec{a} \bullet \vec{b} \right) = 81 4 ∣ a ∣ 2 + b 2 − 4 ( a ∙ b ) = 81 4 ∣ a ⃗ ∣ 2 + ∣ a ⃗ ∣ 2 − 4 ( a ⃗ ∙ b ⃗ ) = 81 4 \left| \vec{a} \right| ^{2} + \left| \vec{a} \right| ^{2} - 4 \left( \vec{a} \bullet \vec{b} \right) = 81 4 ∣ a ∣ 2 + ∣ a ∣ 2 − 4 ( a ∙ b ) = 81 ∵ \because ∵ ∴ \therefore ∴ 5 ∣ a ⃗ ∣ 2 − 4 ( a ⃗ ∙ b ⃗ ) = 81 5 \left| \vec{a} \right| ^{2} - 4 \left( \vec{a} \bullet \vec{b} \right) = 81 5 ∣ a ∣ 2 − 4 ( a ∙ b ) = 81 ⋯ \cdots ⋯ ⋯ \cdots ⋯ a ⃗ ∙ b ⃗ = 1 \vec{a} \bullet \vec{b} = 1 a ∙ b = 1 ∣ a ⃗ ∣ 2 = ∣ b ⃗ ∣ 2 = 17 \left| \vec{a} \right| ^{2} = \left| \vec{b} \right| ^{2} = 17 ∣ a ∣ 2 = b 2 = 17 a ⃗ \vec{a} a b ⃗ \vec{b} b θ \theta θ a ⃗ ∙ b ⃗ = 1 \vec{a} \bullet \vec{b} = 1 a ∙ b = 1 ∣ a ⃗ ∣ ∣ b ⃗ ∣ cos θ = 1 \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta = 1 ∣ a ∣ b cos θ = 1 cos θ = 1 17 \displaystyle \cos \theta = \frac{1}{17} cos θ = 17 1 sin θ = 1 − cos 2 θ = 1 − ( 1 17 ) 2 = 12 2 17 \displaystyle \sin \theta = \sqrt{1 - \cos ^{2} \theta} = \sqrt{1 - \left( \frac{1}{17} \right) ^{2}} = \frac{12 \sqrt{2}}{17} sin θ = 1 − cos 2 θ = 1 − ( 17 1 ) 2 = 17 12 2 O A B \mathrm{OAB} OAB 1 2 ∣ a ⃗ ∣ ∣ b ⃗ ∣ sin θ = 1 2 × 17 × 12 2 17 = 6 2 \displaystyle \frac{1}{2} \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta = \frac{1}{2} \times 17 \times \frac{12 \sqrt{2}}{17} = 6 \sqrt{2} 2 1 ∣ a ∣ b sin θ = 2 1 × 17 × 17 12 2 = 6 2 O A B \mathrm{OAB} OAB a ⃗ \vec{a} a b ⃗ \vec{b} b θ \theta θ O A B \mathrm{OAB} OAB △ \triangle △ O A B \mathrm{OAB} OAB = 1 2 ∣ a ⃗ ∣ ∣ b ⃗ ∣ sin θ \displaystyle = \frac{1}{2} \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta = 2 1 ∣ a ∣ b sin θ = 1 2 ∣ a ⃗ ∣ ∣ b ⃗ ∣ 1 − cos 2 θ \displaystyle = \frac{1}{2} \left| \vec{a} \right| \left| \vec{b} \right| \sqrt{1 - \cos ^{2} \theta} = 2 1 ∣ a ∣ b 1 − cos 2 θ = 1 2 ∣ a ⃗ ∣ ∣ b ⃗ ∣ 1 − ( a ⃗ ∙ b ⃗ ∣ a ⃗ ∣ ∣ b ⃗ ∣ ) 2 \displaystyle = \frac{1}{2} \left| \vec{a} \right| \left| \vec{b} \right| \sqrt{1 - {\left( \frac{{\vec{a} \bullet \vec{b}}}{\left| \vec{a} \right| \left| \vec{b} \right|} \right) ^{2}}} = 2 1 ∣ a ∣ b 1 − ∣ a ∣ b a ∙ b 2 = 1 2 ∣ a ⃗ ∣ 2 ∣ b ⃗ ∣ 2 − ( a ⃗ ∙ b ⃗ ) 2 \displaystyle = \frac{1}{2} \sqrt{\left| \vec{a} \right| ^{2} \left| \vec{b} \right| ^{2} - \left( \vec{a} \bullet \vec{b} \right) ^{2}} = 2 1 ∣ a ∣ 2 b 2 − ( a ∙ b ) 2 = 1 2 17 × 17 − 1 2 \displaystyle = \frac{1}{2} \sqrt{17 \times 17 - 1 ^{2}} = 2 1 17 × 17 − 1 2 = 1 2 ( 17 − 1 ) ( 17 + 1 ) \displaystyle = \frac{1}{2} \sqrt{\left( 17 - 1 \right) \left( 17 + 1 \right)} = 2 1 ( 17 − 1 ) ( 17 + 1 ) = 6 2 \displaystyle = 6 \sqrt{2} = 6 2