Question

If a=3i+4j and b=3j+4k the component of b along a is

Answer 1

Given:

$\vec{a}$ = 3i + 4j = 3i + 4j + 0k

$\vec{b}$ = 3j + 4k = 0i + 3j + 4k

To find:

The component of b along a

Solution:

We will use the following formula to solve the given problem:

$\boxed{\bold{\vec{a}\:.\:\vec{b}}\:=\:|a||b|cos\:\theta}$

where

|a| = length of the vector a

|b| = length of vector b

θ = angle between vector a and vector b

First, we will find the dot product of the two vector a and b,

∴ $\vec{a}\:.\:\vec{b}$

= $(3i + 4j + 0k)\: .\: (0i + 3j + 4k)$

= $(3 \:\times\: 0) + (4 \:\times\:3) + (0 \:\times\: 4)$

= $0 + 12 + 0$

= $12$

Then, we will find the magnitude of $\vec{a}$,

∴ |a|

= $\sqrt{3^2+4^2+0^2}$

= $\sqrt{9 + 16}$

= $\sqrt{25}$

= $5$

Now, using the formula and substituting the values of the dot products of the vectors and the magnitude of vector a, we will find the component of b along a,

∴ |b| cos θ

= $\frac{\vec{a}\:.\:\vec{b}}{|a|}$

= $\frac{12}{5}$

= 2.4

Thus, the component of $\vec{b}$ along $\vec{a}$ is 2.4.

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