2.1 Vectors in the plane  (Page 9/29)

Let $\mathbf{\text{u}}$ and $\mathbf{\text{v}}$ be two nonzero vectors that are nonequivalent. Consider the vectors $\mathbf{\text{a}}=4\mathbf{\text{u}}+5\mathbf{\text{v}}$ and $\mathbf{\text{b}}=\mathbf{\text{u}}+2\mathbf{\text{v}}$ defined in terms of $\mathbf{\text{u}}$ and $\mathbf{\text{v}}.$ Find the scalar $\lambda$ such that vectors $\mathbf{\text{a}}+\lambda \mathbf{\text{b}}$ and $\mathbf{\text{u}}-\mathbf{\text{v}}$ are equivalent.

Let $\mathbf{\text{u}}$ and $\mathbf{\text{v}}$ be two nonzero vectors that are nonequivalent. Consider the vectors $\mathbf{\text{a}}=2\mathbf{\text{u}}-4\mathbf{\text{v}}$ and $\mathbf{\text{b}}=3\mathbf{\text{u}}-7\mathbf{\text{v}}$ defined in terms of $\mathbf{\text{u}}$ and $\mathbf{\text{v}}.$ Find the scalars $\alpha$ and $\beta$ such that vectors $\alpha \mathbf{\text{a}}+\beta \mathbf{\text{b}}$ and $\mathbf{\text{u}}-\mathbf{\text{v}}$ are equivalent.

Consider the vector $\mathbf{\text{a}}(t)=⟨\text{cos}t,\text{sin}t⟩$ with components that depend on a real number $t.$ As the number $t$ varies, the components of $\mathbf{\text{a}}(t)$ change as well, depending on the functions that define them.

1. Write the vectors $\mathbf{\text{a}}(0)$ and $\mathbf{\text{a}}(\pi )$ in component form.
2. Show that the magnitude $\Vert \mathbf{\text{a}}(t)\Vert$ of vector $\mathbf{\text{a}}(t)$ remains constant for any real number $t.$
3. As $t$ varies, show that the terminal point of vector $\mathbf{\text{a}}(t)$ describes a circle centered at the origin of radius $1.$

Consider vector $\mathbf{\text{a}}(x)=⟨x,\sqrt{1-x^2}⟩$ with components that depend on a real number $x\in [\mathrm{-1},1].$ As the number $x$ varies , the components of $\mathbf{\text{a}}(x)$ change as well, depending on the functions that define them.

1. Write the vectors $\mathbf{\text{a}}(0)$ and $\mathbf{\text{a}}(1)$ in component form.
2. Show that the magnitude $\Vert \mathbf{\text{a}}(x)\Vert$ of vector $\mathbf{\text{a}}(x)$ remains constant for any real number $x$
3. As $x$ varies, show that the terminal point of vector $\mathbf{\text{a}}(x)$ describes a circle centered at the origin of radius $1.$

Show that vectors $\mathbf{\text{a}}(t)=⟨\text{cos}t,\text{sin}t⟩$ and $\mathbf{\text{a}}(x)=⟨x,\sqrt{1-x^2}⟩$ are equivalent for $x=r$ and $t=2k\pi ,$ where $k$ is an integer.

Show that vectors $\mathbf{\text{a}}(t)=⟨\text{cos}t,\text{sin}t⟩$ and $\mathbf{\text{a}}(x)=⟨x,\sqrt{1-x^2}⟩$ are opposite for $x=r$ and $t=\pi +2k\pi ,$ where $k$ is an integer.

$\Vert \text{v}\Vert =7,\text{u}=⟨3,4⟩$

$\Vert \text{v}\Vert =3,\text{u}=⟨\mathrm{-2},5⟩$

$\Vert \text{v}\Vert =7,\text{u}=⟨3,\mathrm{-5}⟩$

$\Vert \text{v}\Vert =10,\text{u}=⟨2,\mathrm{-1}⟩$

$\Vert \mathbf{\text{u}}\Vert =2,$ $\theta =30\text{°}$

$\Vert \mathbf{\text{u}}\Vert =6,$ $\theta =60\text{°}$

$\Vert \mathbf{\text{u}}\Vert =5,$ $\theta =\frac{\pi }{2}$

$\Vert \mathbf{\text{u}}\Vert =8,$ $\theta =\pi$

$\Vert \text{u}\Vert =10,$ $\theta =\frac{5\pi }{6}$

$\Vert \text{u}\Vert =50,$ $\theta =\frac{3\pi }{4}$

$\mathbf{\text{u}}=5\sqrt{2}\mathbf{\text{i}}-5\sqrt{2}\mathbf{\text{j}}$

$\mathbf{\text{u}}=\text{−}\sqrt{3}\mathbf{\text{i}}-\mathbf{\text{j}}$

Let $\mathbf{\text{a}}=⟨a_1,a_2⟩,$ $\mathbf{\text{b}}=⟨b_1,b_2⟩,$ and $\mathbf{\text{c}}=⟨c_1,c_2⟩$ be three nonzero vectors. If $a_1{b}_2-a_2{b}_1\ne 0,$ then show there are two scalars, $\alpha$ and $\beta ,$ such that $\mathbf{\text{c}}=\alpha \mathbf{\text{a}}+\beta \mathbf{\text{b}}.$

Consider vectors $\mathbf{\text{a}}=⟨2,\mathrm{-4}⟩,$ $\text{b}=⟨\mathrm{-1},2⟩,$ and 0 Determine the scalars $\alpha$ and $\beta$ such that $\mathbf{\text{c}}=\alpha \mathbf{\text{a}}+\beta \mathbf{\text{b}}.$

Let $P\left(x_0,f\left(x_0\right)\right)$ be a fixed point on the graph of the differential function $f$ with a domain that is the set of real numbers.

1. Determine the real number $z_0$ such that point $Q\left(x_0+1,z_0\right)$ is situated on the line tangent to the graph of $f$ at point $P.$
2. Determine the unit vector $\text{u}$ with initial point $P$ and terminal point $Q.$

Consider the function $f(x)=x^4,$ where $x\in ℝ.$

1. Determine the real number $z_0$ such that point $Q\left(2,z_0\right)$ s situated on the line tangent to the graph of $f$ at point $P\left(1,1\right).$
2. Determine the unit vector $\text{u}$ with initial point $P$ and terminal point $Q.$

Consider $f$ and $g$ two functions defined on the same set of real numbers $D.$ Let $\mathbf{\text{a}}=⟨x,f(x)⟩$ and $\mathbf{\text{b}}=⟨x,g(x)⟩$ be two vectors that describe the graphs of the functions, where $x\in D.$ Show that if the graphs of the functions $f$ and $g$ do not intersect, then the vectors $\text{a}$ and $\text{b}$ are not equivalent.