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The magnitude of the vector is a positive number which represents the length of the vector or directed line segment. It is also known as modulus of the vector. If \(\overrightarrow {OP}\) is the position vector of P, then its magnitude is denoted by \(|\overrightarrow {OP}|\).If \(|\overrightarrow {AB}|\)is a vector, then magnitude is denoted by \(|\overrightarrow {AB}|\). If the vector is written in the single letter from \(\overrightarrow {a}\), then its magnitude is denoted by \(\overrightarrow {a}\).
If the initial point is at the origin and the terminal point at P (x,y).
\(\overrightarrow {OP}\) = \(\frac{x}{y}\) is given by \(|\overrightarrow {OP}|\) = \(\sqrt{(x)^2+(y)^2}\)
If A (x_{1},y_{1}) and B(x_{2},y_{2}) are two points,
We know, \(\overrightarrow {AB}\) =\(\begin{pmatrix} x_2-x_1\\y_2-y_1\end{pmatrix}\). Then \(|\overrightarrow {AB}|\) = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Thus\(|\overrightarrow {AB}|\) = \(\sqrt{(x-component)^2+ (y-component)^2}\)
If the initial point is at A(x_{1},y_{1}) and terminal point at B(x_{2,}y_{2}).
Here, For \(\overrightarrow{AB}\)
x component of \(\overrightarrow{AB}\)= AE = x_{2} - x_{1}
y component of \(\overrightarrow{AB}\) = BE = y_{2} - y_{1}
Using pythagoras theorem,
(AB)^{2} = (AE)^{2} + (BE)^{2}
or, AB^{2} = (x_{2 }- x_{1})^{2} + (y_{2} - y_{1})^{2}
or, AB = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
or, AB = \(\sqrt{(x\;component)^2 + (y\;component)^2}\)
\(\therefore\) \(|\overrightarrow {AB}|\) = AB =\(\sqrt{(x\; comp)^2 + (y\; comp)^2}\)
Direction of a vector is the value of angle made by the vector with x-axis in positive direction. It is generally measured in degree which ranges from 0º to 360º. Direction of a vector is denoted by θ. In the figure, \(\overrightarrow {OP}\) makes an angle θwith x-axis in anticlockwise direction. Draw PM perpendicular to x-axis.
When the initial point is at the origin and terminal point at P(x,y).
Then, OM = x and MP =y.
In ΔPOM, tan θ = \(\frac{Perpendicular}{base}\)
or, tan θ =\(\frac{MP}{OM}\)
or, tan θ = \(\frac{y}{x}\)
∴ θ = tan^{-1 }\((\frac{y}{x}\))
Which gives the direction of the vector.
Hence, direction of vector (θ) =tan^{-1 }\((\frac{y-component}{x-component}\))
When the initial point is at A(x_{1},y_{1}) and the terminal point is at B(x_{2},y_{2})
If A(x_{1},y_{1}) and B(x_{2},y_{2}) are two points, then direction of \(\overrightarrow {AB}\) is given by θ = tan^{-1 }\(\begin{pmatrix}y_2-y_1\\x_2-x_1\\\end{pmatrix}\).
Here, \(\overrightarrow{OA}\) = {3,4}
\(\therefore\) x component = 3 and y component = 4
\(\therefore\) |\(\overrightarrow{OA}\)| = OA = \(\sqrt{(x comp)^2+(y comp)^2}\)
= \(\sqrt{(3)^2 + (4)^2}\)
= \(\sqrt{9+16}\) = \(\sqrt(25)\) = 5
\(\therefore\) \(\overrightarrow{OA}\) = 5 units
Here, P = (\(\sqrt{3}\),\(\sqrt{3}\))
\(\therefore\) x-component = \(\sqrt{3}\)
y - component = \(\sqrt{3}\)
Let \(\theta\) be the angle made by \(\overrightarrow{OP}\) with the positive direction of x-axis.
tan\(\theta\) = \(\frac{y - component}{x - component}\) = \(\frac{\sqrt{3}}{\sqrt{3}}\) = 1
\(\therefore\) tan\(\theta\) = tan45
\(\theta\) = 45 [\(\therefore\) x-component and y-component are +ve the value must lie in 1st quadrant.]
What is the value of column vector if (overrightarrow {AB}) = (egin {pmatrix} 3 \ 4 end {pmatrix}) , (overrightarrow {CD}) = (egin {pmatrix} -4 \ -3 end {pmatrix}) then express (overrightarrow {AB}) + (overrightarrow {CD})?
What is the magnitude of direction if (overrightarrow {AB}) = (egin {pmatrix} 3 \ 4 end {pmatrix}) , (overrightarrow {CD}) = (egin {pmatrix} -4 \ -3 end {pmatrix}) then express (overrightarrow {AB}) + (overrightarrow {CD}) ?
What is the value of column vector if (overrightarrow {PQ}) = (egin {pmatrix} -2 \ 7 end {pmatrix}) and (overrightarrow {RS}) = (egin {pmatrix} 3 \ -2 end {pmatrix}) , then express (overrightarrow {PQ}) + (overrightarrow {RS}) ?
What is the magnitude if (overrightarrow{OA}) = {3,4} ?
What is the direction of (overrightarrow{OP}) where P = ((sqrt{3}),(sqrt{3}))?
What is the column vector of (overrightarrow{AB}) if the vector (overrightarrow{AB}) displaces a point A(2,4) to B(5,7)?
What is the direction of (overrightarrow{AB}) if the vector (overrightarrow{AB}) displaces a point A(2,4) to B(5,7)?
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parallel vector
what happen when two vector are parallel to each other
Mar 24, 2017
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