## Linear Algebra and Its Applications, Exercise 3.2.3

Exercise 3.2.3. Find the multiple of the vector $a = \left(1, 1, 1\right)$ that is closest to the point $b = \left(2, 4, 4\right)$. Also find the point on the line through $b$ that is closest to $a$.

Answer: The first problem amounts to finding the projection $p$ of $b$ onto $a$. From formula (5) on page 147 (theorem 3H) we have $p = \left(a^Tb/a^Ta\right)a$.

Given the definitions of $a$ and $b$ we then have $a^Tb = 1 \cdot 2 + 1 \cdot 4 + 1 \cdot 4 = 2+4+4 = 10$ $a^Ta = 1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1 = 1+1+1 = 3$

We thus have $p = \frac{10}{3}a$ with $\frac{10}{3}$ being the multiple of $a$ we were asked to find.

The second problem amounts to finding the projection $q$ of $a$ onto $b$. Adapting formula (5) we have $q = \left(b^Ta/b^Tb\right)b$.

Given the definitions of $a$ and $b$ we then have $b^Ta = a^Tb = 10$ $b^Tb = 2 \cdot 2 + 4 \cdot 4 + 4 \cdot 4 = 4+16+16 = 36$

We thus have $q = \frac{10}{36}b = \frac{5}{18}b$ or $q = \left(\frac{5}{18} \cdot 2, \frac{5}{18} \cdot 4, \frac{5}{18} \cdot 4\right) = \left(\frac{5}{9}, \frac{10}{9}, \frac{10}{9}\right)$

so that $\left(\frac{5}{9}, \frac{10}{9}, \frac{10}{9}\right)$ is the point on the line through $b$ we were asked to find.

UPDATE: Fixed the computation of $b^Tb$ and thus of $q$. Thanks to Ann and universitypika for the correction.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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### 4 Responses to Linear Algebra and Its Applications, Exercise 3.2.3

1. Ann says:

Hey, for your calculation of bTb I think you miswrote your vector…i got 2*2 + 4*4 + 4*4 = 38.

2. universitypika says:

i got 38 instead of 24 for bTb… 2*2 + 4*4 + 4*4

• universitypika says:

3. hecker says: