Answer to Question #240915 in Physics for Ase

Question #240915

Hello,


The question is the following: the resultant of two vectors, A and B, is a vector of 40 meters oriented towards the north. If the magnitude of A is 30 m and it forms an angle of 30 degrees towards the south in relation to the west, determine the vector of B. Thank you for your help.


1
Expert's answer
2021-09-23T08:30:55-0400

Hello!


First, let's begin with drawing a vector diagram:


Since R is the result of vector sum of A and B (or algebraic sum of their components like AS and AW), and since R does not have neither W- nor E-component, we conclude that the horizontal component of B (BW or BW) must 'cancel' AW. It means that


"A_W+B_x=0,\\\\\nB_x=-A_W=B_E."


Draw this vector:


"B_E=|A_W|=30\\cos30\u00b0=26\\text{ m toward East}."

Consider the vertical component of R. It is a result of subtracting the vertical component of A from the vertical component of B (because A points downward):


"R=B_y-A_y,\\\\\nA_y=A_S,\\\\\nR=B_y-A_S,\\\\\nB_y=R+A_S.\\\\\nB_y=R+A\\sin30\u00b0=40+30\\sin30\u00b0=\\\\=55\\text{ m toward North}."

Draw it:


Therefore, according to Pythagorean theorem, we have


"B=\\sqrt{B_E^2+B_N^2}=61\\text{ m},\\\\\\space\\\\\n\\theta=\\arctan\\frac{B_N}{B_E}=65\u00b0\\text{ East of North}."


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