2.1 Vectors in the plane  (Page 6/29)

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We have found the components of a vector given its initial and terminal points. In some cases, we may only have the magnitude and direction of a vector, not the points. For these vectors, we can identify the horizontal and vertical components using trigonometry ( [link] ).

Consider the angle $\theta$ formed by the vector v and the positive x -axis. We can see from the triangle that the components of vector $\text{v}$ are $⟨\Vert \text{v}\Vert \text{cos}\theta ,\Vert \text{v}\Vert \text{sin}\theta ⟩.$ Therefore, given an angle and the magnitude of a vector, we can use the cosine and sine of the angle to find the components of the vector.

Finding the component form of a vector using trigonometry

Find the component form of a vector with magnitude 4 that forms an angle of $\mathrm{-45}\text{°}$ with the x -axis.

Let $x$ and $y$ represent the components of the vector ( [link] ). Then $x=4\text{cos}\left(\mathrm{-45}\text{°}\right)=2\sqrt{2}$ and $y=4\text{sin}\left(\mathrm{-45}\text{°}\right)=\mathrm{-2}\sqrt{2}.$ The component form of the vector is $⟨2\sqrt{2},\mathrm{-2}\sqrt{2}⟩.$

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Unit vectors

A unit vector    is a vector with magnitude $1.$ For any nonzero vector $\text{v},$ we can use scalar multiplication to find a unit vector $\text{u}$ that has the same direction as $\text{v}.$ To do this, we multiply the vector by the reciprocal of its magnitude:

Recall that when we defined scalar multiplication, we noted that $\Vert k\text{v}\Vert =\left|k\right|·\Vert \text{v}\Vert .$ For $\text{u}=\frac{1}{\Vert \text{v}\Vert }\text{v},$ it follows that $\Vert \text{u}\Vert =\frac{1}{\Vert \text{v}\Vert }\left(\Vert \text{v}\Vert \right)=1.$ We say that $\text{u}$ is the unit vector in the direction of $\text{v}$ ( [link] ). The process of using scalar multiplication to find a unit vector with a given direction is called normalization    .

We have seen how convenient it can be to write a vector in component form. Sometimes, though, it is more convenient to write a vector as a sum of a horizontal vector and a vertical vector. To make this easier, let’s look at standard unit vectors. The standard unit vectors    are the vectors $\text{i}=⟨1,0⟩$ and $\text{j}=⟨0,1⟩$ ( [link] ).

By applying the properties of vectors, it is possible to express any vector in terms of $\text{i}$ and $\text{j}$ in what we call a linear combination :

Thus, $\text{v}$ is the sum of a horizontal vector with magnitude $x,$ and a vertical vector with magnitude $y,$ as in the following figure.