Figure 12.3.4. $\bf V$ is the projection of $\bf A$ onto $\bf B$.
Example 12.3.4 Physical force is a vector quantity. It is often necessary to compute the "component'' of a force acting in a different direction than the force is being applied. For example, suppose a ten pound weight is resting on an inclined plane—a pitched roof, for example. Gravity exerts a force of ten pounds on the object, directed straight down. It is useful to think of the component of this force directed down and parallel to the roof, and the component down and directly into the roof. These forces are the projections of the force vector onto vectors parallel and perpendicular to the roof. Suppose the roof is tilted at a $\ds 30^\circ$ angle, as in figure 12.3.5. A vector parallel to the roof is $\ds \langle-\sqrt3,-1\rangle$, and a vector perpendicular to the roof is $\ds \langle 1,-\sqrt3\rangle$. The force vector is ${\bf F}=\langle 0,-10\rangle$. The component of the force directed down the roof is then $$\eqalign{ {\bf F}_1&={{\bf F}\cdot \langle-\sqrt3,-1\rangle\over|\langle-\sqrt3,-1\rangle|^2} \langle-\sqrt3,-1\rangle ={10\over 2}{\langle-\sqrt3,-1\rangle\over2}= \langle -5\sqrt3/2,-5/2\rangle\cr }$$ with length 5. The component of the force directed into the roof is $$\eqalign{ {\bf F}_2&={{\bf F}\cdot \langle1,-\sqrt3\rangle\over|\langle1,-\sqrt3\rangle|^2} \langle1,-\sqrt3\rangle ={10\sqrt3\over 2}{\langle1,-\sqrt3\rangle\over2}= \langle 5\sqrt3/2,-15/2\rangle\cr }$$ with length $\ds 5\sqrt3$. Thus, a force of 5 pounds is pulling the object down the roof, while a force of $\ds 5\sqrt3$ pounds is pulling the object into the roof. $\square$
Figure 12.3.5. Components of a force. The dot product has some familiar-looking properties that will be useful later, so we list them here. These may be proved by writing the vectors in coordinate form and then performing the indicated calculations; subsequently it can be easier to use the properties instead of calculating with coordinates.
Theorem 12.3.5 If ${\bf u}$, ${\bf v}$, and ${\bf w}$ are vectors and $a$ is a real number, then 1. $\ds {\bf u}\cdot{\bf u} = |{\bf u}|^2$ 2. ${\bf u}\cdot{\bf v} = {\bf v}\cdot{\bf u}$ 3. ${\bf u}\cdot({\bf v}+{\bf w}) = {\bf u}\cdot{\bf v}+{\bf u}\cdot{\bf w}$ 4. $(a{\bf u})\cdot{\bf v}=a({\bf u}\cdot{\bf v}) ={\bf u}\cdot(a{\bf v})$ $\qed$ Exercises 12.3You can use Sage to compute dot products and related quantities like the scalar and vector projections. Ex 12.3.1 Find $\langle 1,1,1\rangle\cdot\langle 2,-3,4\rangle$. (answer) Ex 12.3.2 Find $\langle 1,2,0\rangle\cdot\langle 0,0,57\rangle$. (answer) Ex 12.3.3 Find $\langle 3,2,1\rangle\cdot\langle 0,1,0\rangle$. (answer) Ex 12.3.4 Find $\langle -1,-2,5\rangle\cdot\langle 1,0,-1 \rangle$. (answer) Ex 12.3.5 Find $\langle 3,4,6\rangle\cdot\langle 2,3,4\rangle$. (answer) Ex 12.3.6 Find the cosine of the angle between $\langle 1,2,3\rangle$ and $\langle 1,1,1\rangle$; use a calculator if necessary to find the angle. (answer) Ex 12.3.7 Find the cosine of the angle between $\langle -1, -2,-3\rangle$ and $\langle 5,0,2\rangle$; use a calculator if necessary to find the angle. (answer) Ex 12.3.8 Find the cosine of the angle between $\langle 47,100,0\rangle$ and $\langle 0,0,5\rangle$; use a calculator if necessary to find the angle. (answer) Ex 12.3.9 Find the cosine of the angle between $\langle 1,0,1 \rangle$ and $\langle 0,1,1\rangle$; use a calculator if necessary to find the angle. (answer) Ex 12.3.10 Find the cosine of the angle between $\langle 2,0,0\rangle$ and $\langle -1,1,-1\rangle$; use a calculator if necessary to find the angle. (answer) Ex 12.3.11 Find the angle between the diagonal of a cube and one of the edges adjacent to the diagonal. (answer) Ex 12.3.12 Find the scalar and vector projections of $\langle 1,2,3\rangle$ onto $\langle 1,2,0\rangle$. (answer) Ex 12.3.13 Find the scalar and vector projections of $\langle 1,1,1\rangle$ onto $\langle 3,2,1\rangle$. (answer) Ex 12.3.14 A 20 pound object sits on a ramp at an angle of $\ds 30^\circ$ from the horizontal, as in figure 12.3.5. Find the force pulling the object down the ramp and the force pulling the object directly into the ramp. (answer) Ex 12.3.15 A 20 pound object sits on a ramp at an angle of $\ds 45^\circ$ from the horizontal. Find the force pulling the object down the ramp and the force pulling the object directly into the ramp. (answer) Ex 12.3.16 A force of 10 pounds is applied to a wagon, directed at an angle of $\ds 30^\circ$ from the horizontal, as shown. Find the component of this force pulling the wagon straight up, and the component pulling it horizontally along the ground. (answer)
Figure 12.3.6. Pulling a wagon. Ex 12.3.17 A force of 15 pounds is applied to a wagon, directed at an angle of $\ds 45^\circ$ from the horizontal. Find the component of this force pulling the wagon straight up, and the component pulling it horizontally along the ground. (answer) Ex 12.3.18 A force ${\bf F}$ is to be applied to a wagon, directed at an angle of $\ds 30^\circ$ from the horizontal. The resulting force pulling the wagon horizontally along the ground is to be 10 pounds. What is the magnitude of the required force ${\bf F}$? (answer) Ex 12.3.19 Use the dot product to find a non-zero vector ${\bf w}$ perpendicular to both ${\bf u}=\langle 1,2,-3\rangle$ and ${\bf v}=\langle 2,0,1\rangle$. (answer) Ex 12.3.20 Let ${\bf x}=\langle 1,1,0 \rangle$ and ${\bf y}=\langle 2,4,2 \rangle$. Find a unit vector that is perpendicular to both $\bf x$ and $\bf y$. (answer) Ex 12.3.21 Do the three points $(1,2,0)$, $(-2,1,1)$, and $(0,3,-1)$ form a right triangle? (answer) Ex 12.3.22 Do the three points $(1,1,1)$, $(2,3,2)$, and $(5,0,-1)$ form a right triangle? (answer) Ex 12.3.23 Show that $|{\bf A}\cdot{\bf B}|\le|{\bf A}||{\bf B}|$ Ex 12.3.24 Let $\bf x$ and $\bf y$ be perpendicular vectors. Use Theorem 12.3.5 to prove that $\ds |{\bf x}|^2+|{\bf y}|^2=|{\bf x}+{\bf y}|^2$. What is this result better known as? Ex 12.3.25 Prove that the diagonals of a rhombus intersect at right angles. Ex 12.3.26 Suppose that ${\bf z}=|{\bf x}| {\bf y} + |{\bf y}| {\bf x}$ where $\bf x$, $\bf y$, and $\bf z$ are all nonzero vectors. Prove that $\bf z$ bisects the angle between $\bf x$ and $\bf y$. Ex 12.3.27 Prove Theorem 12.3.5. Let the angle between \[\vec{a}\] and \[\vec{b}\] be \[\theta\] It is given that \[\left| \vec{a} \right| = \left| \vec{b} \right| = \left| \sqrt{3} \vec{a} - \vec{b} \right| = 1\] \[\left| \sqrt{3} \vec{a} - \vec{b} \right| = 1\]\[ \Rightarrow \left| \sqrt{3} \vec{a} - \vec{b} \right|^2 = 1\]\[ \Rightarrow \left| \sqrt{3} \vec{a} \right|^2 - 2\sqrt{3} \vec{a} . \vec{b} + \left| \vec{b} \right|^2 = 1\] \[ \Rightarrow 3 \left| \vec{a} \right|^2 - 2\sqrt{3}\left| \vec{a} \right|\left| \vec{b} \right|\cos\theta + \left| \vec{b} \right|^2 = 1\] \[\Rightarrow 3 \times 1 - 2\sqrt{3} \times 1 \times 1 \times \cos\theta + 1 = 1\] \[ \Rightarrow 2\sqrt{3}\cos\theta = 3\] \[ \Rightarrow \cos\theta = \frac{\sqrt{3}}{2} = \cos\frac{\pi}{6}\] \[ \Rightarrow \theta = \frac{\pi}{6}\] Thus, the angle between \[\vec{a}\] and \[\vec{b}\] is \[\frac{\pi}{6}\] |
If a and b are unit vectors then what is the angle between a and b for √2a-b to be a unit vector
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