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Here's a question from my homework:

Two vectors are given by a = 1.9i + 3.2j and b = 1.6i + 8.6j. Find (a)|a × b|, (b)a · b, (c)(a + b) · b, and (d) the component of a along the direction of b?

I'm just fine with a-d, but I don't know what they're asking for in (d). Any ideas what this problem is wanting?

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I would think that they want you to write $\vec{a}$ as a sum of two vectors: one parallel to $\vec{b}$ (the one they want as an answer) and the other orthogonal to $\vec{b}$. – Jyrki Lahtonen Sep 9 '11 at 3:43
up vote 2 down vote accepted

Yes. They are asking you to compute what is the length of $a$ after it has been projected on the axis that goes through the vector $b$.

Another way to state this : If you draw your vectors $\vec a$ and $\vec b$ on the plane, you can choose a vector that is orthogonal to $\vec b = (1.6,8.6)$, say for instance $\vec c = (-8.6,1.6)$ because in general a vector of the form $(x,y)$ is orthogonal to the vector $(-y,x)$ (it suffices to compute the scalar product to confirm that). After that, you want to add a certain multiple of this orthogonal vector to $\vec a$, to ensure that $\vec a + \lambda \vec c$ is parallel to $\vec b$, i.e, that you projected orthogonally $\vec a$ on $\vec b$. Say that this is done, i.e. that $$ \vec a + \lambda \vec c = \mu \vec b $$ then by computing the scalar product by $\vec b$ on both sides, you obtain $$ \vec b \cdot \vec a = \vec b \cdot \vec a + \lambda (0) = \vec b \cdot \vec a + \lambda (\vec b \cdot \vec c)= \vec b \cdot (\vec a + \lambda \vec c) = \mu \vec b \cdot \vec b \quad \Longrightarrow \quad \mu = \frac{\vec b \cdot \vec a}{\vec b \cdot \vec b}. $$ because by construction $\vec b \cdot \vec c = 0$. Now the component of $\vec a$ along the direction of $\vec b$ is precisely $$ \mu \| \vec b \| = \frac{\vec b \cdot \vec a}{\vec b \cdot \vec b} \sqrt{ \vec b \cdot \vec b} = \frac{\vec b \cdot \vec a}{\sqrt{\vec b \cdot \vec b}}. $$ Hope that helps!

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Thank you for your help! – BigBoy Sep 9 '11 at 13:08

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