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- Question 9 Exercise 3.4
- we have,\\ \begin{align}\vec{c} \times \vec{d}&=\left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \... ghtarrow{ } \\ & =\dfrac{3}{2} \hat{i}+\hat{j}-k+\left(\dfrac{1}{2} \hat{i}-\dfrac{3}{2} \hat{j}+2 \hat{... we have,\\ \begin{align}\vec{c} \times \vec{d}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... ay}\right|\\ \Rightarrow \vec{c} \times \vec{d}&=\left(\dfrac{5}{2}-\dfrac{3}{2}\right) \hat{i}-(1+6) \h
- Question 2 Exercise 3.4
- = First Way \begin{align}\vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... \sqrt{56}}\\ \Rightarrow \quad \theta&=\cos ^{-1}\left(\frac{-28}{2 \sqrt{14} \sqrt{14}}\right) \\ \Righ... = First Way \begin{align}\vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... {14} \sqrt{126}} \\ \Rightarrow \theta&=\cos^{-1}\left(\sqrt{\dfrac{42 \times 42}{14 \times 126}}\right)
- Question 3 & 4 Exercise 3.5
- egin{align}\vec{a} \cdot \vec{b} \times \vec{c}&=\left|\begin{array}{ccc} 3 & 0 & 2 \\ 1 & 2 & 1 \\ 0 & ... . .(1) \\ \vec{b} \cdot \vec{c} \times \vec{a}&=\left|\begin{array}{ccc} 1 & 2 & 1 \\ 0 & -1 & 4 \\ 3 &... . .(2) \\ \vec{c} \cdot \vec{a} \times \vec{b}&=\left|\begin{array}{ccc} 0 & -1 & 4 \\ 3 & 0 & 2 \\ 1 &... vec{b} \\ \text { Now } \vec{c} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i}& \hat{j} & \hat{k} \\
- Question 7 & 8 Exercise 3.3
- dot product \begin{align}\vec{a} \cdot \vec{b}&=\left(-\dfrac{3}{2} \hat{j}+\dfrac{4}{5} \hat{k}\right)... t{k}) \\ \Rightarrow \vec{a} \cdot \vec{b}&=0(1)+\left(-\dfrac{3}{2}\right)(-2)+\dfrac{4}{5}(-2) \\ \Rig... \ldots . .(1) \\ \text { Also }|\vec{a}|&=\sqrt{\left(-\dfrac{3}{2}\right)^2+\left(\dfrac{4}{5}\right)^2} \\ \Rightarrow|\vec{a}|&=\sqrt{\dfrac{9}{4}+\dfrac{
- Question 3 Exercise 3.4
- vec{b}} \\ \text { Now } \vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... dots(1) \\ \text { Now } \vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... ts \\ \Rightarrow|\vec{a} \times \vec{b}|&=\sqrt{\left(-\left.25\right|^2+(3)^2+(13)^2\right.} \\ \Rightarrow|\vec{a} \times \vec{b}|&=\sqrt{803} \ldots \ldot
- Question 2 and 3 Exercise 3.3
- cdot \sqrt{29}} \\ \Rightarrow \theta&=\cos ^{-1}\left(\dfrac{8}{5 \cdot \sqrt{29}}\right) \\ \Rightarro... qrt{13} . \\ \text { and }|\vec{b}|&=\sqrt{(1)^2+\left(1^{12}+(1)^2\right.} \\ \Rightarrow|\vec{b}|&=\sq... cdot \sqrt{13}} \\ \Rightarrow \theta&=\cos ^{-1}\left(\dfrac{-1}{\sqrt{3} \cdot \sqrt{13}}\right) \\ \R... imately)}\end{align} ====Go To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit03:ex3
- Question 4 Exercise 3.4
- olution==== \begin{align}\vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... olution==== \begin{align}\vec{b} \times \vec{c}&=\left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \... 2) \\ (\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... hat{k} .\end{align} ====Go To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit03:ex3
- Question 7 & 8 Exercise 3.4
- dots \ldots \ldots(1) \\ \vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... ec{a} \times \vec{b}} \\ \vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... c} =10$ then\\ $\vec{c}= \vec{c} \cdot \hat{n}=10\left(\dfrac{10 \hat{i}+12 \hat{j}+4 \hat{k}}{\sqrt{65}... e required vector.\\ ====Go To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit03:ex3
- Question 5(i) & 5(ii) Exercise 3.5
- egin{align}\vec{a} \cdot \vec{a} \times \vec{b}&=\left|\begin{array}{lll} a_1 & a_2 & a_3 \\ a_1 & a_2 &... egin{align}\vec{b} \cdot \vec{a} \times \vec{b}&=\left|\begin{array}{lll} b_1 & b_2 & b_3 \\ a_1 & a_2 &... We know that \begin{align}\vec{a}\times \vec{b}&=\left|\begin{array}{lll} \hat{i}&\hat{j}&\hat{k}\\ a_1 ... b_2-a_2 b_1)^2.$$ ====Go To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit03:ex3
- Question 7 Exercise 3.5
- {u} \cdot \vec{v} \times \vec{w}&=0\\ \Rightarrow\left|\begin{array}{ccc}1 & 2 & 3 \\ 2 & -3 & 4 \\ 3 & ... {u} \cdot \vec{v} \times \vec{w}&=0\\ \Rightarrow\left|\begin{array}{ccc} 1 & 1 & -1 \\ 1 & -2 & 1 \\ c ... u} \cdot \bar{v} \times \vec{w}&=0 \\ \Rightarrow\left|\begin{array}{lll} 1 & 1 & 2 \\ 2 & 3 & 1 \\ c & ... become coplanar. ====Go To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit03:ex3
- Question 1 Exercise 3.4
- \vec{b}&=\hat{j} \times(2 \hat{j}+3 \hat{k})\\ &=\left|\begin{array}{lll}\hat{i} & \hat{j} & \hat{k} \\ ... }&=\hat{k} \\ \therefore \vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... -3 \hat{k} \\ \therefore \vec{a} \times \vec{b}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\
- Question 5 Exercise 3.4
- \overrightarrow{PQ}\times \overrightarrow{P R}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... overrightarrow{P Q} \times \overrightarrow{P R}&=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\... square. }\end{align} ====Go To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit03:ex3
- Question 6 Exercise 3.4
- \text { Hence } \vec{M}-\vec{r} \times \vec{F}&=\left|\begin{array}{ccc} i & j & k \\ 1 & -2 & 2 \\ 3 &... (0 ,-4,1) \\ \vec{M}&=\vec{r} \times \vec{F}\\ &=\left |\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \... t{k} .\end{align} ====Go To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit03:ex3
- Question 8 Exercise 3.5
- {v} \times \vec{w}]\\ \Rightarrow V&=\dfrac{1}{6}\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 0 & 7... of tetiahedron is: \begin{align}V&=\dfrac{1}{6} \left| \begin{array}{rrr} -3 & -5 & -1 \\ -2 & -1 & -6 ... }{6}$ units cube. ====Go To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit03:ex3
- Question 9 Exercise 3.5
- n{align} (\hat{i} \times \hat{j}) \cdot \hat{k}&=\left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & ... n{align} (\hat{k} \times \hat{j}) \cdot \hat{i}&=\left|\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & ... 1+0=-1 \text {. }$$ ====Go To==== <text align="left"><btn type="primary">[[math-11-kpk:sol:unit03:ex3