The magnitude of the vector [tex](\vec A+\vec B)[/tex] is [tex]\boxed{16 \text{ units}}[/tex] and the direction of the vector [tex](\vec A+\vec B)[/tex] is [tex]\boxed{26.56^\circ\text{ anticlockwise}}[/tex]
Further Explanation:
[tex]\Vec A[/tex] points in the negative Y-direction and has a magnitude of [tex]7\text{ units}[/tex], so we will show it as a line of length [tex]7\text{ units}[/tex] in negative Y-direction.
[tex]\Vec B[/tex] has twice the magnitude and points in the positive X-direction, so we will show it by a line of double the length as before in positive X-direction, as shown in the figure below.
According to Cartesian coordinate system, the resultant will start either from tail of [tex]\vec A[/tex] and ends at head of [tex]\vec B[/tex] and vice-versa.
Now, resultant will be given by
[tex]\begin{aligned}\vec R&=\vec A+\vec B\\\vec R&=({-7\hat j+14\hat i})\\\end{aligned}[/tex]
Hence, the magnitude of resultant or of vector [tex]\vec A + \vec B[/tex] is given by:
[tex]\begin{aligned}|{\vec R}|&=\sqrt{({7^2}+{14^2})}\\&=\sqrt{(49+196)}\\&=\sqrt{245}\\&=15.65{\text{ units}}\\\end{aligned}[/tex]
It can be expressed in whole number as:
[tex]\boxed{|\vec R|=16\text{ units}}[/tex]
Now, the direction of the resultant vector [tex]\vec R[/tex] or of the vector [tex]\vec A + \vec B[/tex] is given by:
[tex]\tan\theta=\dfrac{{|{\vec A}|}}{{|{\vec B}|}}[/tex]
Now, substituting the values of vector [tex]\vec A[/tex] and [tex]\vec B[/tex]:
[tex]\begin{aligned}\tan\theta&=\dfrac{7}{{14}}\\\tan\theta&=0.5\\\theta&={\tan^{-1}}(0.5)\\&=\boxed{26.56^\circ}\\\end{aligned}[/tex]
Thus, the magnitude of the vector [tex](\vec A+\vec B)[/tex] is [tex]\boxed{16 \text{ units}}[/tex] and the direction of the vector [tex](\vec A+\vec B)[/tex] is [tex]\boxed{26.56^\circ\text{ anticlockwise}}[/tex]
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Answer Details:
Grade: High School
Subject: Physics
Chapter: Vectors and Scalars
Keywords:
Vector, scalars, vector A, Vector B, addition, resultant, unit, magnitude, direction, positive, X-axis, Y- axis, negative, 7 units, 14 units, twice the magnitude.
