Question

How do you find the vector v with the given magnitude of 9 and in the same direction as \[u = < 2,5 > \]?

Answer 1

Hint:
The key point to find the vector is to use Pythagoras theorem, which states that the square of the hypotenuse is equal to the sum of squares of the other two sides, and here with respect to the given magnitude and direction we can find the vectors of the given equation.

Complete step by step solution:
To find the vector v with the given magnitude, let us consider the direction given as \[u1 = 2\], \[u2 = 5\].
As we know vector a with given magnitude in the direction of b, is given by:
\[\dfrac{{\parallel a\parallel \cdot b}}{{\parallel b\parallel }}\]
We can use Pythagoras theorem as,
\[\parallel u\parallel = \sqrt {{{\left( {u1} \right)}^2} + {{\left( {u2} \right)}^2}} \]
\[\parallel u\parallel = \sqrt {{{\left( 2 \right)}^2} + {{\left( 5 \right)}^2}} \]
After simplifying the terms, we get
\[\parallel u\parallel = \sqrt {4 + 25} \]
\[\parallel u\parallel = \sqrt {29} \]
As the given magnitude is
\[\parallel v\parallel = 9\]
Substituting these values in the formula as,
\[\dfrac{{\parallel a\parallel \cdot b}}{{\parallel b\parallel }}\]
= \[\dfrac{{9\left( {2 + 5} \right)}}{{\sqrt {29} }}\]
Hence, the vector is
= \[\dfrac{{18}}{{\sqrt {29} }}\mathop i\limits^ \wedge + \dfrac{{45}}{{\sqrt {29} }}\mathop j\limits^ \wedge \]
Therefore, the vectors are:
\[\dfrac{{18}}{{\sqrt {29} }}\] and \[\dfrac{{45}}{{\sqrt {29} }}\]

Additional information:
Vector is an object which has magnitude and direction both. Magnitude defines the size of the vector. It is represented by a line with an arrow, where the length of the line is the magnitude of the vector and the arrow shows the direction. It is also known as Euclidean vector or Geometric vector or Spatial vector or simply “vector “.
The magnitude of a vector is shown by vertical lines on both the sides of the given vector. It represents the length of the vector. Mathematically, the magnitude of a vector is calculated by the help of “Pythagoras Theorem,”

Formula used:
\[\parallel u\parallel = \sqrt {{{\left( {u_1} \right)}^2} + {{\left( {u_2} \right)}^2}} \]
Where, u1 and u2 are the directions of the given vector.

Note:
The magnitude of a vector gives the length of the line segment, while the direction gives the angle the line forms with the positive x-axis. Two vectors are said to equal if their magnitude and direction are the same. According to vector algebra, a vector can be added to another vector.