Chapter 9: Q. 38 (page 613)

Show that the projection of vonto iis $(v \cdot i) i .$ Then show that we can always write a vector vas
$v = (v \cdot i) i + (v \cdot j) j$

Short Answer

Expert verified

Yes, the projection ofvontoiis $(v \cdot i) i$ and the vectorvis $(v \cdot i) i + (v \cdot j) j .$

Step by step solution

Step 1. Given Information

We have to show that the projection of vonto iis $(v \cdot i) i$ and vectorvis $(v \cdot i) i + (v \cdot j) j .$

Step 2. Showing the projection of v onto i

As we know If vand ware two nonzero vectors, the vector projection of vonto w is

$v_{1} = \frac{v \cdot w}{{||w||}^{2}} w$

iis the unit vector, so the projection of vonto iis

role="math" localid="1646824336209" $v_{1} = \frac{v \cdot i}{{||i||}^{2}} i v_{1} = (v \cdot i) i [{||i||}^{2} = 1] . . . . . . . . . . (a)$

Hence proved that the projection ofvontoiis $(v \cdot i) i .$

Step 3. Showing the vector v

We have found the projection of vonto i,now let's find the projection of vonto j.

jis the unit vector so,

$v_{2} = \frac{v \cdot j}{{||j||}^{2}} j v_{2} = (v \cdot j) j [{||j||}^{2} = 1] . . . . . . . . . (b)$

We can write vector vas

$v = v_{1} + v_{2} v = (v \cdot i) i + (v \cdot j) j$

Hence proved.

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Show that the projection of vonto iis $(v \cdot i) i .$ Then show that we can always write a vector vas
$v = (v \cdot i) i + (v \cdot j) j$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Showing the projection of v onto i

Step 3. Showing the vector v

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Show that the projection of vonto iis v·ii. Then show that we can always write a vector vasv=(v·i)i+(v·j)j

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Showing the projection of v onto i

Step 3. Showing the vector v

One App. One Place for Learning.

Most popular questions from this chapter

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Show that the projection of vonto iis $(v \cdot i) i .$ Then show that we can always write a vector vas
$v = (v \cdot i) i + (v \cdot j) j$