Maria starts at the origin and walks along all of the vector $2 extbf{i} + 3 extbf{j}$, then walks along all of the vector $3 extbf{i} - 2 extbf{j}$ and finally along all of the vector $4 extbf{i} - 3 extbf{j}$ - HSC - SSCE Mathematics Extension 1 - Question 4 - 2020 - Paper 1
Question 4
Maria starts at the origin and walks along all of the vector $2 extbf{i} + 3 extbf{j}$, then walks along all of the vector $3 extbf{i} - 2 extbf{j}$ and finally alon... show full transcript
Worked Solution & Example Answer:Maria starts at the origin and walks along all of the vector $2 extbf{i} + 3 extbf{j}$, then walks along all of the vector $3 extbf{i} - 2 extbf{j}$ and finally along all of the vector $4 extbf{i} - 3 extbf{j}$ - HSC - SSCE Mathematics Extension 1 - Question 4 - 2020 - Paper 1
Step 1
Calculate the resultant vector after walking along $2\textbf{i} + 3\textbf{j}$
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Answer
The first vector is ( \textbf{A} = 2\textbf{i} + 3\textbf{j} ). After moving along this vector, Maria's position becomes ( \textbf{R_1} = \textbf{A} = (2, 3) ).
Step 2
Calculate the resultant vector after walking along $3\textbf{i} - 2\textbf{j}$
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Answer
The second vector is ( \textbf{B} = 3\textbf{i} - 2\textbf{j} ). Now adding this to the current position: ( \textbf{R_2} = \textbf{R_1} + \textbf{B} = (2 + 3, 3 - 2) = (5, 1) ).
Step 3
Calculate the resultant vector after walking along $4\textbf{i} - 3\textbf{j}$
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Answer
The third vector is ( \textbf{C} = 4\textbf{i} - 3\textbf{j} ). Now adding this to the current position: ( \textbf{R_3} = \textbf{R_2} + \textbf{C} = (5 + 4, 1 - 3) = (9, -2) ).
Step 4
Determine the final distance from the origin
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Answer
To find the distance from the origin to the final position ( \textbf{R_3} = (9, -2) ), we use the distance formula:
d=(x2+y2)=(92+(−2)2)=(81+4)=85.
Thus, the distance from the origin is ( \sqrt{85} ).
