Using properties of determinants prove that:
Question: Using properties of determinants prove that:. $\left|\begin{array}{ccc}(\mathrm{m}+\mathrm{n})^{2} \mathrm{l}^{2} \mathrm{mn} \\ (\mathrm{n}+1)^{2} \mathrm{~m}^{2} \ln \\ (1+\mathrm{m})^{2} \mathrm{n}^{2} \operatorname{lm}\end{array}\right|=\left(\mathrm{l}^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}\right)(1-\mathrm{m})$ $(m-n)(n-1)$ Solution:...
Read More →Let A_1 A_2 A_3 A_4 A_5 A_6 A_1 be a regular hexagon.
Question: Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{1}$ be a regular hexagon. Write the x-components of the vectors represented by the six sides taken in order. Use the fact that the resultant of these six vectors is zero, to prove that $\cos 0+\cos \pi / 3+\cos 2 \pi / 3+\cos 4 \pi / 3+\cos 5 \pi / 3=0$ Use the known cosine values to verify the result. Solution: From polygon law of vector addition, the resultant of the six vectors can be affirmed to be zero. Here their magnitudes are the same...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}(x-2)^{2} (x-1)^{2} x^{2} \\ (x-1)^{2} x^{2} (x+1)^{2} \\ x^{2} (x+1)^{2} (x+2)^{2}\end{array}\right|=-8$ Solution:...
Read More →Two vectors have magnitudes
Question: Two vectors have magnitudes $2 \mathrm{~m}$ and $3 \mathrm{~m}$. The angle between them is $60^{\circ}$. Find (a) the scalar product of the two vectors, (b) the magnitude of their vector product? Solution: We have $a=2 m, b=3 m$. $\Theta=60^{\circ}$ is the angle between the two vectors Scalar product between the two vectors $=a \cdot b=$ $2 \times 3 \times \cos \left(60^{\circ}\right)=3 \mathrm{~m}^{2}$ Vector product between the two vectors $=a \times b=$ $2 \times 3 \times \sin \left...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}a^{2} b^{2} c^{2} \\ (a+1)^{2} (b+1)^{2} (c+1)^{2} \\ (a-1)^{2} (b-1)^{2} (c-1)^{2}\end{array}\right|=4(a-b)(b-c)(c-a)$ Solution:...
Read More →Prove the following
Question: Suppose $\vec{a}$ is a vector of magnitude $4.5$ unit due north. What is the vector (a) $3 \vec{a}$ (b) $-4 \vec{a}$ ? Solution: $a=4.5 \mathbf{n}$, where $\mathbf{n}$ is unit vector in north direction (a) $3 a=4.5 \times 3 n=13.5$ in north direction (b) $-4 a=4.5 X-4 n=18$ in south direction...
Read More →A mosquito net over a
Question: A mosquito net over a $7 \mathrm{ft}^{\times} 4 \mathrm{ft}$ bed is $3 \mathrm{ft}$ high. The net has a hole at one corner of the bed through which a mosquito enters the bed. It flies and sits at the diagonally opposite upper corner of the net. (a) Find the magnitude of the displacement of the mosquito. (b) Taking the hole as the origin, the length of the bed as the X-axis, its width as the $Y$-axis, and vertically up as the Z-axis, write the components of the displacement vector. Solu...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}a b a x+b y \\ b c b x+c y \\ a x+b y b x+c y 0\end{array}\right|=\left(b^{2}-a c\right)\left(a x^{2}+3 b x y+c y^{2}\right)$ Solution:...
Read More →A carrom board
Question: A carrom board ( $4 \mathrm{ft}^{\times} 4 \mathrm{ft}$ square) has the queen at the center. The queen, hit by the striker moves to the front edge, rebounds and goes in the hole behind the striking line. Find the magnitude of displacement of the queen (a) from the center to the front edge, (b) from the front edge to the hole and (c) from the center to the hole. Solution: In $\otimes \mathrm{ABC}, \tan (=x / 2$ and in $\otimes \mathrm{DCF}, \tan (=(2-\mathrm{x}) / 4$, So, $(\mathrm{x} /...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}a+b+c -c -b \\ -c a+b+c -a \\ -b -a a+b+c\end{array}\right|=2(a+b)(b+c)(c+a)$ Solution:...
Read More →A spy report about a suspected car reads as follows.
Question: A spy report about a suspected car reads as follows. "The car moved $2.00 \mathrm{~km}$ towards east, made a perpendicular left turn, ran for $500 \mathrm{~m}$, made a perpendicular right turn, ran for $4.00 \mathrm{~km}$ and stopped." Find the displacement of the car. Solution: $A B=2 i+0.5 j+4 i=6 i+0.5 j$ As the car went forward, took a left and then a right. So, $\mathrm{AB}=\left(6^{2}+0.5^{2}\right)^{1 / 2}=6.02 \mathrm{~km}$ And $\phi=\tan ^{-1}(\mathrm{BE} \backslash \mathrm{AE...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: Solution:...
Read More →Two vectors have magnitudes 3 unit and 4 unit respectively.
Question: Two vectors have magnitudes 3 unit and 4 unit respectively. What should be the angle between them if the magnitude of the resultant is (a) 1 unit, (b) 5 unit and (c) 7 unit. Solution: $|a|=3$ and $|b|=4$ Let $\theta$ be the angle between them. Then, using the relation $R^{2}=A^{2}+B^{2}+2 A B \cos \theta$ (a) We get for $R=1$, $1=9+16+24 \operatorname{Cos} \theta$ Or, $\theta=180^{\circ}$ (b) For, $\mathrm{R}=5$, we have $25=9+16+24 \operatorname{Cos} \theta$ Or, $\cos \theta=0$; $\the...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}b+c a a \\ b c+a b \\ c c a+b\end{array}\right|=4 a b c$ Solution:...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{lll}b+c a-b a \\ c+a b-c b \\ a+b c-a c\end{array}\right|=3 a b c-a^{3}-b^{3}-c^{3}$ Solution:...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}x y z \\ x^{2} y^{2} z^{2} \\ x^{3} y^{3} z^{3}\end{array}\right|=x y z(x-y)(y-z)(z-x)$ Solution:...
Read More →Refer to the figure. (a) Find the magnitude,
Question: Refer to the figure. (a) Find the magnitude, (b) $x$ and $y$ components and (c) the angle with the $\mathrm{X}$-axis of the resultant of $\overrightarrow{O A}, \overrightarrow{B C}$ and $\overrightarrow{D E}$. Solution: $x$ component of $\mathrm{OA}=2 \cos 30^{\circ}=\sqrt{3}$ $x$ component of $B C=1.5 \cos 120^{\circ}=-0.75$ $x$ component of $D E=1 \cos 270^{\circ}=0$ $y$ component of $\mathrm{OA}=2 \sin 30^{\circ}=1$ component of $B C=1.5 \sin 120^{\circ}=1.3$ component of $\mathrm{D...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}3 x -x+y -x+z \\ x-y 3 y z-y \\ x-z y-z 3 z\end{array}\right|=3(x+y+z)(x y+y z+z x)$ Solution: $=(x+y+z)(3 x y+3 y z+3 x z)$ $=3(x+y+z)(x y+y z+z x)$...
Read More →Solve this following
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}x x+y x+2 y \\ x+2 y x x+y \\ x+y x+2 y x\end{array}\right|=9 y^{2}(x+y)$ Solution:...
Read More →Prove the following
Question: Let $\vec{a}=4 \hat{\imath}+3 \hat{\jmath}$ and $\vec{b}=3 \hat{\imath}+4 \hat{\jmath}$. (a) Find the magnitudes of (a) $\vec{a}$, (b) $\vec{b}$, (c) $\vec{a}+\vec{b}$ and (d) $\vec{a}-\vec{b}$. Solution: $a=4 i+3 j, b=3 i+4 j$ $|a|=|b|=\sqrt{\left(3^{2}+4^{2}\right)}=5$ $a+b=7 i+7 j$ and $a-b=i-j$ $|a+b|$ $=\sqrt{\left(7^{2}+7^{2}\right)}=7 \sqrt{2}$ and $|a-b|=\sqrt{\left(1^{2}+1^{2}\right)}=\sqrt{2}$...
Read More →Add vectors vector A, vector B and vector C each having magnitude of
Question: Add vectors $\vec{A}, \vec{B}$ and $\vec{C}$ each having magnitude of 100 unit and inclined to the $\mathrm{X}$-axis at angles $45^{\circ}, 135^{\circ}$ and $315^{\circ}$ respectively. Solution: Vectors $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are oriented at $45^{\circ}, 135^{\circ}$ and $315^{\circ}$ respectively. $|A|=|B|=|C|=100$ units Let $A=A_{x} \mathbf{i}+A_{y} \mathbf{j}+A_{z} \mathbf{k}, B=B_{x} \mathbf{i}+B_{y} \mathbf{j}+B_{z} \mathbf{k}$, and $C=C_{x} \mathbf{i}+C_{y} \ma...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}a^{2}+2 a 2 a+1 1 \\ 2 a+1 a+2 1 \\ 3 3 1\end{array}\right|=(a-1)^{3}$ Solution:...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}x+\lambda 2 x 2 x \\ 2 x x+\lambda 2 x \\ 2 x 2 x x+\lambda\end{array}\right|=(5 x+\lambda)(\lambda-x)^{2}$ Solution:...
Read More →Let vector A and vector B be the two vectors of magnitude 10 unit each.
Question: Let $\vec{A}$ and $\vec{B}$ be the two vectors of magnitude 10 unit each. If they are inclined to the $\mathrm{X}$-axis at angles $30^{\circ}$ and $60^{\circ}$ respectively, find the resultant. Solution: $A$ and $B$ are inclined at angles of 30 degrees and 60 degrees with respect to the $x$ axis Angle between them $=(60-30)=90$ degrees Given that $|\mathrm{A}|=|\mathrm{B}|=10$ units, we get $\mathrm{R}^{2}=\mathrm{A}^{2}+\mathrm{B}^{2}+2 \mathrm{AB} \cos \theta$ $=10^{2}+10^{2}+2.10 .1...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}x+4 2 x 2 x \\ 2 x x+4 2 x \\ 2 x 2 x x+4\end{array}\right|=(5 x+4)(x-4)^{2}$ Solution:...
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