3-44 할리데이 11판 솔루션 일반물리학 {r⃗=(5.25i^+4.90j^+3.00k^)[m] \begin{cases} \vec r &= \(5.25\i+4.90\j+3.00\k\)\ut{m} \end{cases} {r=(5.25i^+4.90j^+3.00k^)[m] (a)\ab{a}(a) r=5.252+4.92+32=12024229[m]≈7.782833674183202[m]≈7.78[m] \begin{aligned} r &= \sqrt{5.25^2+4.9^2+3^2}\\ &=\frac{1}{20}\sqrt{24229}\ut{m}\\ &\approx 7.782833674183202\ut{m}\\ &\approx 7.78\ut{m}\\ \end{aligned} r=5.252+4.92+32=20124229[m]≈7.782833674183202[m]≈7.78[m] (b)\ab{b}(b) l<r ?l\lt r ~?l<r ? [Impossible]\title{Impossible}[Impossible] (c)\ab{c}(c) l>r ?l\gt r ~?l>r ? [Possible]\title{Possible}[Possible] (d)\ab{d}(d) l=r ?l = r ~?l=r ? [Possible]\title{Possible}[Possible] (e)\ab{e}(e) $$\ve.. 11판/3. 벡터 2024.01.25
3-43 할리데이 11판 솔루션 일반물리학 {A⃗=7.00[km]j^B=10.5[km]θB=60° \begin{cases} \vec A &= 7.00\ut{km}\j\\ B&=10.5\ut{km}\\ \theta_B&=60\degree\\ \end{cases} ⎩⎨⎧ABθB=7.00[km]j^=10.5[km]=60° B⃗=10.5cos60°i^+10.5sin60°j^=214i^+2143j^ \begin{aligned} \vec B &=10.5\cos60\degree\i+10.5\sin60\degree\j\\ &=\frac{21}{4}\i+\frac{21}{4}\sqrt{3}\j \end{aligned} B=10.5cos60°i^+10.5sin60°j^=421i^+4213j^ C⃗=−B⃗+A⃗=−(214i^+2143j^)+7j^=−214i^+(7−2143)j^ \begin{aligned} \vec C &=-\vec B+\vec A\\ &=-\(\frac{21}{4}\i+\frac{21}{4}\sqrt{3}\j\)+7\j\\ &=-\frac{21}{4}\i+\(7-\frac{21}{4}\sqrt{3}\)\j\\ \end{aligned} C=−B+A=−(421i^+4213j^)+7j^=−421i^+(7−4213)j^ (a)\ab{a}(a) $.. 11판/3. 벡터 2024.01.25
3-42 할리데이 11판 솔루션 일반물리학 {d⃗=−1.5[m]j^ \begin{cases} \vec d &=-1.5\ut{m}\j \end{cases} {d=−1.5[m]j^ (a)\ab{a}(a) ∣6.0d⃗∣=∣−9j^∣=9 \begin{aligned} \abs{6.0\vec d} &= \abs{-9\j}\\ &=9 \end{aligned} 6.0d=−9j^=9 (b)\ab{b}(b) South\text{South}South (c)\ab{c}(c) ∣−2.0d⃗∣=∣3j^∣=3 \begin{aligned} \abs{-2.0\vec d} &= \abs{3\j}\\ &=3 \end{aligned} −2.0d=3j^=3 (b)\ab{b}(b) North\text{North}North 11판/3. 벡터 2024.01.25
3-41 할리데이 11판 솔루션 일반물리학 {a⃗=6.0i^+5.0j^b⃗=3.0i^+4.0j^ \begin{cases} \vec a&=6.0\i+5.0\j\\ \vec b&=3.0\i+4.0\j\\ \end{cases} {ab=6.0i^+5.0j^=3.0i^+4.0j^ (a)\ab{a}(a) a⃗×b⃗=(aybz−byaz)i^+(azbx−bzax)j^+(axby−bxay)k^, \vec a\times \vec b=(a_yb_z-b_ya_z)\i+(a_zb_x-b_za_x)\j+(a_xb_y-b_xa_y)\k, a×b=(aybz−byaz)i^+(azbx−bzax)j^+(axby−bxay)k^, a⃗×b⃗=(6i^+5j^)×(3i^+4j^)=(6×4−5×3)k^=9k^ \begin{aligned} \vec a \times \vec b &=\(6\i+5\j\)\times\(3\i+4\j\)\\ &=(6\times4-5\times3)\k\\ &=9\k \end{aligned} a×b=(6i^+5j^)×(3i^+4j^)=(6×4−5×3)k^=9k^ (b)\ab{b}(b) a⃗⋅b⃗=axbx+ayby+azbz,\vec a \cdot \vec b=a_xb_x+a_yb_y+a_zb_z,a⋅b=axbx+ayby+azbz, $$ \begin{aligned} \vec a \cdot \vec b&=\(6\i+5\j\).. 11판/3. 벡터 2024.01.25
3-40 할리데이 11판 솔루션 일반물리학 {East:+i^North:+j^up:+k^ \begin{cases} \text{East}:+\i\\ \text{North}:+\j\\ \text{up}:+\k\\ \end{cases} ⎩⎨⎧East:+i^North:+j^up:+k^ (a)\ab{a}(a) Ans=(−j^)×(−i^)=(−1)(−1)(j^×i^)=(j^×i^)=(−1)(i^×j^)=(−1)(k^)=−k^Down \begin{aligned} \Ans&=\(-\j\)\times\(-\i\)\\ &=(-1)(-1)\(\j\times\i\)\\ &=\(\j\times\i\)\\ &=(-1)\(\i\times\j\)\\ &=(-1)\(\k\)\\ &=-\k\\ &\text{Down} \end{aligned} Ans=(−j^)×(−i^)=(−1)(−1)(j^×i^)=(j^×i^)=(−1)(i^×j^)=(−1)(k^)=−k^Down (b)\ab{b}(b) $$ \begin{aligned} \Ans&=\(-\k\)\times\(-\j\)\\ &=(-1)(-1)\(\k\times\j\)\\ &=\(\k\times\j\)\\ &=(-1)\(\j\times\k\)\\ &=(-.. 11판/3. 벡터 2024.01.25
3-39 할리데이 11판 솔루션 일반물리학 {A⃗+B⃗=1.0i^+6.0j^A⃗−B⃗=−5.0i^+2.0j^ \begin{cases} \vec A + \vec B &=1.0\i+6.0\j\\ \vec A - \vec B &=-5.0\i+2.0\j\\ \end{cases} {A+BA−B=1.0i^+6.0j^=−5.0i^+2.0j^ 2A⃗=−4.0i^+8.0j^A⃗=−2.0i^+4.0j^ \begin{aligned} 2\vec A&=-4.0\i+8.0\j\\ \vec A&=-2.0\i+4.0\j\\ \end{aligned} 2AA=−4.0i^+8.0j^=−2.0i^+4.0j^ 11판/3. 벡터 2024.01.25
3-38 할리데이 11판 솔루션 일반물리학 {d⃗1=(−2.0i^+3.0j^+4.0k^)[m]d⃗2=(−2.0i^−4.0j^+3.0k^)[m]d⃗3=(2.0i^+3.0j^+2.5k^)[m] \begin{cases} \vec d_1 &= \(-2.0\i+3.0\j+4.0\k\)\ut{m}\\ \vec d_2 &= \(-2.0\i-4.0\j+3.0\k\)\ut{m}\\ \vec d_3 &= \(2.0\i+3.0\j+2.5\k\)\ut{m}\\ \end{cases} ⎩⎨⎧d1d2d3=(−2.0i^+3.0j^+4.0k^)[m]=(−2.0i^−4.0j^+3.0k^)[m]=(2.0i^+3.0j^+2.5k^)[m] (a)\ab{a}(a) a⃗⋅b⃗=axbx+ayby+azbz,\vec a \cdot \vec b=a_xb_x+a_yb_y+a_zb_z,a⋅b=axbx+ayby+azbz, $$ \begin{aligned} \Ans&=\vec d_1\cdot\(\vec d_2+\vec d_3\)\\ &=\vec d_1\cdot\bra{\(-2\i-4\j+3\k\)+\(2\i+3\j+2.5\k\)}\\ &=\(-2\i+3\j+4\k\)\cdot\(-\j+5.5\k\)\\ &=(-2)\cdot0+3\cdot(-1.. 11판/3. 벡터 2024.01.25
3-37 할리데이 11판 솔루션 일반물리학 {a⃗=8i^b⃗=6j^c⃗=−8i^−6j^ \begin{cases} \vec a &= 8\i\\ \vec b &= 6\j\\ \vec c &= -8\i-6\j\\ \end{cases} ⎩⎨⎧abc=8i^=6j^=−8i^−6j^ (a)\ab{a}(a) a⃗⋅b⃗=8⋅0+0⋅6=0 \begin{aligned} \vec a\cdot\vec b&=8\cdot0+0\cdot6\\ &=0 \end{aligned} a⋅b=8⋅0+0⋅6=0 (b)\ab{b}(b) a⃗⋅c⃗=8⋅(−8)+0⋅(−6)=−64 \begin{aligned} \vec a\cdot\vec c&=8\cdot(-8)+0\cdot(-6)\\ &=-64 \end{aligned} a⋅c=8⋅(−8)+0⋅(−6)=−64 (c)\ab{c}(c) b⃗⋅c⃗=0⋅(−8)+6⋅(−6)=−36 \begin{aligned} \vec b\cdot\vec c&=0\cdot(-8)+6\cdot(-6)\\ &=-36 \end{aligned} b⋅c=0⋅(−8)+6⋅(−6)=−36 11판/3. 벡터 2024.01.25
3-36 할리데이 11판 솔루션 일반물리학 {A⃗=0B⃗=55[km]i^r1=24[km],θ1=−25°r⃗2=8.0[km]j^ \begin{cases} \vec A &= 0\\ \vec B &= 55\ut{km}\i\\ r_1&=24\ut{km}, \theta_1=-25\degree\\ \vec r_2 &= 8.0\ut{km}\j \end{cases} ⎩⎨⎧ABr1r2=0=55[km]i^=24[km],θ1=−25°=8.0[km]j^ C⃗=(A⃗+r⃗1+r⃗2)−B⃗=(0+24cos(−25°)i^+24sin(−25°)+8j^)−55i^=(24cos25°−55)i^+(8−24sin25°)j^ \begin{aligned} \vec C &= \(\vec A + \vec r_1 + \vec r_2\)-\vec B\\ &=\(0 + 24\cos(-25\degree)\i+24\sin(-25\degree) + 8\j\)-55\i\\ &=(24\cos25\degree-55)\i+(8-24\sin25\degree)\j\\ \end{aligned} C=(A+r1+r2)−B=(0+24cos(−25°)i^+24sin(−25°)+8j^)−55i^=(24cos25°−55)i^+(8−24sin25°)j^ $$ \begin{aligned} C &= \sqrt{(24\cos25\degree-55)^2+(8-24\sin25\.. 11판/3. 벡터 2024.01.25
3-35 할리데이 11판 솔루션 일반물리학 {a⃗=ai^b⃗=−bj^d>0 \begin{cases} \vec a &= a\i\\ \vec b &= -b\j\\ d&>0 \end{cases} ⎩⎨⎧abd=ai^=−bj^>0 (a)\ab{a}(a) b⃗d=−bj^d=−bdj^ \begin{aligned} \frac{\vec b}{d}&=\frac{-b\j}{d}\\ &=-\frac{b}{d}\j\\ \end{aligned} db=d−bj^=−dbj^ −bd<0-\frac{b}{d}\lt0−db<0 ∴−y\therefore -y∴−y (b)\ab{b}(b) b⃗−d=−bj^−d=bdj^ \begin{aligned} \frac{\vec b}{-d}&=\frac{-b\j}{-d}\\ &=\frac{b}{d}\j\\ \end{aligned} −db=−d−bj^=dbj^ bd>0\frac{b}{d}\gt0db>0 ∴+y\therefore +y∴+y (c)\ab{c}(c) $$ \begin{aligned} \vec a \cdot \vec .. 11판/3. 벡터 2024.01.24