What is the projection of $(j + 2 k)$ $(j+2k)$ onto $(i + 2 j)$ $( i+2j)$ ?

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What is the projection of $(j + 2 k)$ $(j+2k)$ onto $(i + 2 j)$ $( i+2j)$ ?

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Answer 1 · 2016-12-07T07:07:24

Scalar projection is $\frac{2}{\sqrt{5}}$ $2/sqrt5$

and vector projection is $\frac{2}{5} j + \frac{4}{5} k$ $2/5j+4/5k$

Explanation:

Projection of a vector $\to a$ $veca$ onto another vector $\to b$ $vecb$ could be scalar projection or vector projection.

A scalar projection is $\frac{\to a \cdot \to b}{∣ ∣ \to a ∣ ∣}$ $(veca*vecb)/|veca|$ and vector projection is $\frac{\to a \cdot \to b}{\to a \cdot \to a} \cdot \to a$ $(veca*vecb)/(veca*veca)*veca$ , where $\to a$ $veca$ is one on whom we are seeking projection

Here as $\to b = (i + 2 j)$ $vecb=(i+2j)$ and $\to a = (j + 2 k)$ $veca=(j+2k)$ , hence

$\to a \cdot \to b = 1 \times 0 + 2 \times 1 + 0 \times 2 = 2$ $veca*vecb=1xx0+2xx1+0xx2=2$

and $\to a \cdot \to a = 1 \times 1 + 2 \times 2 = 5$ $veca*veca=1xx1+2xx2=5$ and hence $∣ ∣ \to a ∣ ∣ = \sqrt{5}$ $|veca|=sqrt5$

Hence scalar projection is $\frac{2}{\sqrt{5}}$ $2/sqrt5$

and vector projection is $\frac{2}{5} \times (j + 2 k) = \frac{2}{5} j + \frac{4}{5} k$ $2/5xx(j+2k)=2/5j+4/5k$

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What is the projection of $(j + 2 k)$ $(j+2k)$ onto $(i + 2 j)$ $( i+2j)$ ?

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