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July 3, 2017

Vector Analysis

Let's discuss a beautiful problem useful for Physics Olympiad based on Vector Analysis.

Vector Analysis Problem:

Let (\vec{a}=6\vec{i}-3\vec{j}-6\vec{k}) and (\vec{d}=\vec{i}+\vec{j}+\vec{k}). Suppose that (\vec{a}=\vec{b}+\vec{c}) where (\vec{b}) is parallel to (\vec{d}) and (\vec{c}) is perpendicular to (\vec{d}). Then (\vec{c}) is
(B)   ( 7\vec{i}-2\vec{j}-5\vec{k})
(C)    (4\vec{i}-5\vec{j}+\vec{k})
(D)    (3\vec{i}+6\vec{j}-9\vec{k})


In the given problem, (\vec{a})=(6\vec{i}-3\vec{j}-6\vec{k})
$$\vec{d}=\vec{i}+\vec{j}+\vec{k}$$ and $$\vec{a}=\vec{b}+\vec{c}...(i)$$
Now, let us consider (\vec{b}=\lambda\vec{d}) and (\vec{c}).(\vec{d})=)0.
Therefore, (b=\lambda\vec{i}+\lambda\vec{j}+\lambda\vec{j})
From (1),

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