2.3 The dot product  (Page 8/16)

[T] $\text{a}=⟨3,\mathrm{-1}⟩,$ $\mathbf{\text{b}}=⟨\mathrm{-4},0⟩$

[T] $\text{a}=⟨2,1⟩,$ $\mathbf{\text{b}}=⟨\mathrm{-1},3⟩$

$\text{u}=3\text{i},$ $\text{v}=4\text{i}+4\text{j}$

$\text{u}=5\mathbf{\text{i}},$ $\text{v}=\mathrm{-6}\text{i}+6\text{j}$

$\text{a}=⟨3,\mathrm{-1},2⟩,$ $\mathbf{\text{b}}=⟨1,\mathrm{-1},\mathrm{-2}⟩$

$\text{a}=⟨0,\mathrm{-1},\mathrm{-3}⟩,$ $\mathbf{\text{b}}=⟨2,3,\mathrm{-1}⟩$

$\text{a}=\text{i}+\text{j},$ $\mathbf{\text{b}}=\text{j}-\mathbf{\text{k}}$

$\text{a}=\text{i}-2\mathbf{\text{j}}+\text{k},$ $\mathbf{\text{b}}=\text{i}+\text{j}-2\text{k}$

[T] $\text{a}=3\text{i}-\text{j}-2\text{k},$ $\mathbf{\text{b}}=\text{v}+\mathbf{\text{w}},$ where $\text{v}=\mathrm{-2}\text{i}-3\text{j}+2\text{k}$ and $\text{w}=\text{i}+2\text{k}$

[T] $\text{a}=3\text{i}-\text{j}+2\text{k},$ $\mathbf{\text{b}}=\text{v}-\mathbf{\text{w}},$ where $\text{v}=2\text{i}+\text{j}+4\text{k}$ and $\text{w}=6\text{i}+\text{j}+2\text{k}$

$\text{a}=⟨x,y⟩,$ $\mathbf{\text{b}}=⟨\text{−}y,x⟩,$ where x and y are nonzero real numbers

$\text{a}=⟨x,x⟩,$ $\mathbf{\text{b}}=⟨\text{−}y,y⟩,$ where x and y are nonzero real numbers

$\text{a}=3\text{i}-\text{j}-2\text{k},$ $\mathbf{\text{b}}=\mathrm{-2}\text{i}-3\mathbf{\text{j}}+\mathbf{\text{k}}$

$\text{a}=\text{i}-\text{j},$ $\mathbf{\text{b}}=7\text{i}+2\mathbf{\text{j}}-\mathbf{\text{k}}$

Find all two-dimensional vectors a orthogonal to vector $\mathbf{\text{b}}=⟨3,4⟩.$ Express the answer in component form.

Find all two-dimensional vectors a orthogonal to vector $\mathbf{\text{b}}=⟨5,\mathrm{-6}⟩.$ Express the answer by using standard unit vectors.

Determine all three-dimensional vectors u orthogonal to vector $\text{v}=⟨1,1,0⟩.$ Express the answer by using standard unit vectors.

Determine all three-dimensional vectors u orthogonal to vector $\text{v}=\text{i}-\text{j}-\mathbf{\text{k}}.$ Express the answer in component form.

Determine the real number $\alpha$ such that vectors $\text{a}=2\text{i}+3\mathbf{\text{j}}$ and $\mathbf{\text{b}}=9\text{i}+\alpha \mathbf{\text{j}}$ are orthogonal.

Determine the real number $\alpha$ such that vectors $\text{a}=\mathrm{-3}\text{i}+2\mathbf{\text{j}}$ and $\mathbf{\text{b}}=2\text{i}+\alpha \mathbf{\text{j}}$ are orthogonal.

[T] Consider the points $P(4,5)$ and $Q(5,\mathrm{-7}).$

1. Determine vectors $\overrightarrow{OP}$ and $\overrightarrow{OQ}.$ Express the answer by using standard unit vectors.
2. Determine the measure of angle O in triangle OPQ . Express the answer in degrees rounded to two decimal places.

[T] Consider points $A(1,1),$ $B(2,\mathrm{-7}),$ and $C(6,3).$

1. Determine vectors $\overrightarrow{BA}$ and $\overrightarrow{BC}.$ Express the answer in component form.
2. Determine the measure of angle B in triangle ABC . Express the answer in degrees rounded to two decimal places.

Determine the measure of angle A in triangle ABC , where $A(1,1,8),$ $B(4,\mathrm{-3},\mathrm{-4}),$ and $C(\mathrm{-3},1,5).$ Express your answer in degrees rounded to two decimal places.

Consider points $P(3,7,\mathrm{-2})$ and $Q(1,1,\mathrm{-3}).$ Determine the angle between vectors $\overrightarrow{OP}$ and $\overrightarrow{OQ}.$ Express the answer in degrees rounded to two decimal places.

$\text{u}=⟨3,7,\mathrm{-2}⟩,$ $\text{v}=⟨5,\mathrm{-3},\mathrm{-3}⟩,$ $\text{w}=⟨0,1,\mathrm{-1}⟩$

$\text{u}=\text{i}-\text{k},$ $\text{v}=5\mathbf{\text{j}}-5\text{k},$ $\text{w}=10\mathbf{\text{j}}$

Use vectors to show that a parallelogram with equal diagonals is a square.

Use vectors to show that the diagonals of a rhombus are perpendicular.

Show that $\text{u}·(\text{v}+\text{w})=\text{u}·\text{v}+\text{u}·\text{w}$ is true for any vectors u , v , and w .

Verify the identity $\text{u}·(\text{v}+\text{w})=\text{u}·\text{v}+\text{u}·\text{w}$ for vectors $\text{u}=⟨1,0,4⟩,$ $\text{v}=⟨\mathrm{-2},3,5⟩,$ and $\text{w}=⟨4,\mathrm{-2},6⟩.$