Checkpoint 1.2.1.
Calculate the distance between \(A\) and \(B.\)
Section 1.2 Vector Operations
Subsection 1.2.1 Distance
There are a variety of applications when we want to know the distance between two points (locations of two objects). Using other properties in this section we can also learn to calculate the distance between points and lines (like an object and a wall), between parallel lines, between points and planes, and similar questions.
Use these points to practice calculating distances between points. Let \(A=(2,3),\) \(B=(1,-5),\) \(\vec{u}=(2,3),\) \(\vec{v}=(1,8),\) \(X=(5,4,7),\) \(Y=(1,3,-4),\) \(\vec{w}=(4,1,11).\)
Checkpoint 1.2.2.
Calculate the distance between \(X\) and \(Y\text{.}\) Your instincts for this are most likely correct.
The notation we use for distance derives from the absolute value symbol. Recall with numbers the absolute value could be used to determine the distance between two numbers. For example \(|5-3|=2,\) so 5 and 3 are 2 units apart. The distance between two points is written
\begin{equation*}
\|X-Y\|.
\end{equation*}
Subsection 1.2.2 Magnitude
Just as the absolute value gives us the distance between two numbers it gives the magnitude of any one number. Namely, the absolute value is the distance of a number from 0 on the number line or, simply stated, how big the number is without concern for which direction from 0. This idea is used to define the magnitude of a vector. Because a vector can be calculated as the distance between two points (one at tail and the other at the head), the distance calculation for two points is also the magnitude of a vector.
Definition 1.2.3. Magnitude.
The magnitude of a vector \(\vec{u}\text{,}\) denoted \(\|\vec{u}\|\text{,}\) is the distance between the origin and a point at the end of the vector (placed at the origin).
Use the points and vectors from the previous problem sets for the following calculations.
Checkpoint 1.2.4.
How far is \(A\) from the origin?
Checkpoint 1.2.5.
Calculate \(\|\vec{u}\|\text{.}\)
Checkpoint 1.2.6.
Calculate \(\|\stackrel{\longrightarrow}{AB}\|\text{.}\)
Checkpoint 1.2.7.
Calculate \(\|\vec{w}\|\text{.}\)
Checkpoint 1.2.8.
Calculate \(\|\stackrel{\longrightarrow}{XY}\|\text{.}\)
Subsection 1.2.3 Unit Vectors
Recall that a vector has both magnitude and direction. Sometimes it is useful to remove the magnitude. We do this by scaling the magnitude of the vector so that it is 1.
Example 1.2.9.
Consider \(\vec{u}=(8,2,3)\text{.}\) The magnitude is \(\|\vec{u}\|=\sqrt{8^2+2^2+3^2}=\sqrt{77}\text{.}\) The vector in the same direction with unit length is therefore
\begin{equation*}
\frac{1}{\sqrt{77}}\vec{u}=\left(\frac{8}{\sqrt{77}}, \frac{2}{\sqrt{77}}, \frac{3}{\sqrt{77}} \right)
\end{equation*}
You can confirm this by calculating the magnitude of this vector.
Note unit vectors are often denoted as \(\hat{i}=\langle 1,0,0 \rangle\) or \(\hat{u}=\left\langle\frac{8}{\sqrt{77}}, \frac{2}{\sqrt{77}}, \frac{3}{\sqrt{77}} \right\rangle\text{.}\)
Checkpoint 1.2.10.
Find the unit vector in the direction of \(\vec{v}=(1,17,23)\text{.}\)
Checkpoint 1.2.11.
Find the unit vector in the direction of \(\vec{v}=-7\vec{i}+8\vec{j}-9\vec{k}\text{.}\)
Subsection 1.2.4 Spheres
Definition 1.2.12. Circle.
A circle is a curve (2D) consisting of the set points such that they are all equidistant from a fixed point called the center.
Definition 1.2.13. Sphere.
A sphere is the surface (2D in 3D) consisting of the set of points such that they are all equidistant from a fixed point called the center.
The coordinate planes are the \(xy\)-plane, \(xz\)-plane, and the \(yz\)-plane.
Activity 5.
First derive an equation for a sphere using point and vector notations.
(a)
Let \(P\) be a point on the sphere and \(C\) be the fixed point that is the center. Write an expression for the distance between \(P\) and \(C\text{.}\)
(b)
Let the fixed distance between points \(P\) on the sphere and the center \(C\) be the positive, real value \(r.\) Write an equation for a sphere using point/vector notation.
(c)
In this notation is there any difference between the equation for a circle and a sphere?
(d)
Let \(P=(x,y,z)\) and \(C=(c_x,c_y,c_z)\) and the radius be \(r\text{.}\) Write the equation for the sphere using this notation.
(e)
Write the equation for a sphere with center \(C=(4,7,-5)\) and radius 5. Describe the intersections of this sphere with the coordinate planes.
(f)
Write the equation for a sphere with center \(C=(1,3,2)\) that passes through the origin.
Subsection 1.2.5 Dot Product
Definition 1.2.14. Dot Product.
The dot product of two vectors \(\vec{u}=(u_1,u_2,\ldots,u_n)\) and \(\vec{v}=(v_1,v_2,\ldots,v_n)\text{,}\) denoted \(\vec{u} \cdot \vec{v},\) is the scalar \(u_1v_1+u_2v_2+\ldots+u_nv_n.\)
Activity 6.
Note the dot product is a function from a pair of vectors to a scalar. While this function may appear new, this activity shows us that it is connected to the cosine function we already know.
(a)
Write the ratio for \(\cos \theta\) in terms of right triangle \(ABO.\)
(b)
Write \(\mbox{ratio}=\cos \theta\text{.}\)
(c)
Re-write \(\theta\) in terms of \(\alpha\) and \(\beta.\)
(d)
Re-write the \(\cos \theta\) using an appropriate trig identity.
(e)
Replace each trig function from the identity with its appropriate ratio in terms of appropriate triangles. You may wish to sketch lines perpendicular to the \(x\)-axis from \(A\) and from \(B.\)
(f)
Simplify.
(g)
Identify the dot product in this equation and solve for it.
(h)
Note the lengths of the vectors are in the equation. Use the standard magnitude notation.
Instructions.
Investigate the effect of angle on the dot product by rotating the two angles. Investigate the effect of scale on the dot product by scale the two vectors.
Activity 7.
This relationship between the dot product and the cosine function implies a connection between the dot product and angles. In this activity we determine the relationship between the dot product and the geometric properties of orthogonality and parallelism.
Use the following below. \(\vec{a}_1=(1,1,1),\) \(\vec{b}_1=(1,1,-2),\) and \(\vec{b}_2=(2,2,2).\)
(a)
Sketch \(\ell_2(t)=(1,3,-2)+\vec{a}_1 t.\)
(b)
Sketch \(\ell_3(t)=(1,3,-2)+\vec{b}_1 t.\)
(c)
Sketch \(\ell_4(t)=(1,3,-2)+\vec{b}_2 t.\)
(d)
Calculate \(\vec{a}_1 \cdot \vec{b}_1.\)
(e)
Calculate \(\vec{a}_1 \cdot \vec{b}_2.\)
(f)
Compare and contrast \(\ell_2(t)\) and \(\ell_3(t).\)
(g)
Compare and contrast \(\ell_2(t)\) and \(\ell_4(t).\)
(h)
If \(\vec{a} \cdot \vec{b} = 0,\) what is the (geometric) relationship between the two vectors?
(i)
If \(\vec{a} \cdot \vec{b} = \|\vec{a}\|\|\vec{b}\|,\) what is the (geometric) relationship between the two vectors?
(j)
If \(0 \lt \vec{a} \cdot \vec{b} \lt \|\vec{a}\|\|\vec{b}\|,\) what is the relationship?
Theorem 1.2.17.
For any two non-zero vectors \(\vec{u}\) and \(\vec{v}\) the angle \(\theta\) between them satisfies
\begin{equation*}
\cos(\theta) = \frac{\vec{u}}{\|\vec{u}\|} \cdot \frac{\vec{v}}{\|\vec{v}\|}
\end{equation*}
Note this states that the cosine of the angle is the dot product of the unitized vectors. Think for a moment why the angle between two vectors in 3D, 4D, or 29D makes sense.
Subsection 1.2.6 Orthogonal Projection
The following definition is based on the trigonometric relationship discovered above.
Definition 1.2.18. Orthogonal Projection.
The orthogonal projection of \(\vec{x}\) on \(\vec{y}\) is given by \(\mbox{proj}_{\vec{y}} \vec{x} = \frac{\vec{x} \cdot \vec{y}}{\| \vec{y} \|} \cdot \frac{\vec{y}}{\|\vec{y}\|}. \)
Instructions.
Investigate the orthogonal projection by moving points B and C (changing the scale of the vectors and angle between) and seeing the effect on the projection.
Activity 8.
We note that the projection is orthogonal. In this activity we take advantage of this to break down a pair of vectors into orthogonal parts.
For the following use \(\vec{x}=(5,1)\) and \(\vec{y}=(1,3).\)
(a)
Calculate \(\vec{z}=\mbox{proj}_{\vec{y}} \vec{x}.\)
(b)
Sketch \(\vec{x}, \vec{y},\) and \(\vec{z}.\)
(c)
Describe the relationship between \(\vec{x}, \vec{y},\) and \(\vec{z}.\)
(d)
Calculate the new vector \(\vec{z}^\prime = \vec{x}-\vec{z}.\)
(e)
Describe the relationship between \(\vec{y}, \vec{z},\) and \(\vec{z}^\prime.\)
This idea of decomposing a set of vectors into a set of orthogonal vectors is illustrated in Figure 1.2.19. The vector \(\vec{v}=\stackrel{\longrightarrow}{AC}\) is decomposed into the orthogonal pair of vectors: \(\stackrel{\longrightarrow}{AD}\) and \(\stackrel{\longrightarrow}{DC}\text{.}\) It will be extended in linear algebra.
For the following use \(\vec{a}=(6,1,2),\) \(\vec{b}=(7,6,2),\) \(\vec{c}=(1,2,-4).\)
Checkpoint 1.2.20.
Determine if any pair of these three vectors are orthogonal.
Checkpoint 1.2.21.
Calculate \(\mbox{proj}_{\vec{a}} \vec{b}\) and a vector orthogonal to it.
Checkpoint 1.2.22.
Calculate \(\mbox{proj}_{\vec{a}} \vec{c}\) and a vector orthogonal to it.
Subsection 1.2.7 Dot Product Properties
Activity 9.
Discover and list properties of the dot product from each of these experiments.
(a)
Let \(\vec{a}=(a_1,a_2,a_3)\) and \(\vec{b}=(b_1,b_2,b_3).\)
(i)
Expand \(|\vec{a} \cdot \vec{b}|.\)
(ii)
Expand \(\|\vec{a}\| \|\vec{b}\|.\)
(iii)
Compare the two results and write an inequality.
(b)
Choose some \(\vec{a}\) and \(\vec{b}\) (probably in 2D). Draw \(\vec{a},\) and \(\vec{a}+\vec{b}\) from the origin and \(\vec{b}\) from the head of \(\vec{a}.\) (This is the usual demonstration of vector addition.) What is the relationship between \(\|\vec{a}\|^2+\|\vec{b}\|^2\) and \(\|\vec{a}+\vec{b}\|^2\text{?}\)
(c)
Now find a case such that \(\|\vec{a}\|^2+\|\vec{b}\|^2=\|\vec{a}+\vec{b}\|^2\)
(d)
Find specific vectors such that \(\vec{a} \cdot \vec{b}= \vec{a} \cdot \vec{c} \) but \(\vec{b} \ne \vec{c}\) and \(\vec{a} \ne \vec{0}.\)
Subsection 1.2.8 Cross Product
Definition 1.2.23. Cross Product.
The cross product of two vectors \(\vec{u}=(u_1,u_2,u_3)\) and \(\vec{v}=(v_1,v_2,v_3)\text{,}\) denoted \(\vec{u} \times \vec{v},\) is the vector \((u_2v_3-u_3v_2,-(u_1v_3-u_3v_1),u_1v_2-u_2v_1).\)
Note the cross product is a function from a pair of vectors to a vector.
Activity 10.
This activity displays properties of the cross product.
Use \(\vec{a}=(6,1,2),\) \(\vec{b}=(7,6,2),\) \(\vec{c}=(1,2,-4).\)
(a)
These tasks demonstrate the geometric relationship of the cross product to the original vectors
(i)
Calculate \(\vec{d}=\vec{a} \times \vec{b}.\)
(ii)
Calculate \(\vec{a} \cdot \vec{d}.\)
(iii)
Calculate \(\mbox{proj}_{\vec{b}} \vec{d}.\)
(b)
These tasks illustrate the direction of the cross product.
(i)
Calculate \(\vec{f_1}=\vec{a} \times \vec{c}.\)
(ii)
Calculate \(\vec{f_2}=\vec{c} \times \vec{a}.\)
(c)
How many vectors are pairwise orthogonal to two vectors in 3D? 4D?
Activity 11.
This activity investigates the question: how many orthogonal vectors are there?
For the following use \(\vec{x}=(-1,2,-1),\) \(\vec{y}=(3,3,-2),\) \(\vec{z}=(2,1,0).\)
(a)
Determine if any pair of these vectors are orthogonal.
(b)
Produce a vector orthogonal to \(\vec{x}\) and \(\vec{y}.\)
(c)
Produce a vector orthogonal to \(\vec{z}\) using the following method. Write \(\vec{z} \cdot (u_1,u_2,u_3)=0\text{.}\) Perform the dot product then find a solution by selecting two values and solving for the third.
(d)
How many pairs of vectors are orthogonal to \(\vec{z}\text{?}\)
(e)
If \(\vec{a} \times \vec{b} = \vec{a} \times \vec{c}\) and \(\vec{a} \ne \vec{0}\) must \(\vec{b}=\vec{c}\text{?}\)
