Vectors in R2 and R3 | Free Calculus 2 Lesson (4 College Credits) | UPI Study

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Vectors in R2 and R3

Calculus 2 · Chapter 10: Vectors & the Geometry of SpaceSection 10.1 · lesson 51 of 548 min · free

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In this lesson

Quantities that carry a direction.

  • Compute components and magnitude
  • Normalise a vector to extract pure direction
  • Add, subtract and scale, algebraically and geometrically
  • Write vectors in standard basis form

Transcript

Everything said in the video, section by section (1,062 words). Read along, or search it.

MAT-265 - Calculus 2 · Section 10.1

Vectors in Two and Three Dimensions

Temperature, mass and time are fully described by a single number. Those are scalars. But a wind of twenty knots is meaningless to a pilot without a bearing. And a fifty-newton push does very different things depending on which way it points. Quantities like those need magnitude and direction together. And the mathematical object that carries both is the vector.

The Object

Magnitude and Direction Together

A vector is represented geometrically by a directed line segment. An arrow. Its components are obtained by subtracting the initial coordinates from the terminal ones. So a vector from P one to P two has components x two minus x one, y two minus y one, z two minus z one. And we write those in angle brackets — which distinguishes a vector from a point, which uses parentheses. The magnitude, or length, is the square root of the sum of the squared components. Now here's the property that defines what a vector really is. A vector is unchanged by translation. Only its components matter. Not where it's drawn. So an arrow from the origin to three, four, two — and an arrow from one, one, one, to four, five, three — are the same vector. Because both have components three, four, two. When a vector is positioned with its initial point at the origin, it's said to be in standard position. And then its terminal point has coordinates matching its components exactly.

Direction Alone

Unit Vectors and the Standard Basis

Sometimes you want direction without magnitude. Dividing a vector by its own length produces a unit vector. One of magnitude exactly one. It carries the direction, with all information about size stripped away. Let's normalize the vector four, three, twelve — say it's a drone's displacement. First find the magnitude. Root of sixteen plus nine plus one forty-four. Root one sixty-nine. Thirteen. Divide each component by thirteen. You get four thirteenths, three thirteenths, twelve thirteenths. Verify the length is one. Square each component: sixteen over one sixty-nine, plus nine over one sixty-nine, plus one forty-four over one sixty-nine. That's one sixty-nine over one sixty-nine. One. Confirmed. And this vector encodes the drone's heading, with no reference to distance. Multiplying it by any positive number yields a vector along the identical path. Useful when the same route must be flown at different speeds, or scaled to a different range. There's also a standard set of three unit vectors, pointing along the axes. i is one, zero, zero. j is zero, one, zero. And k is zero, zero, one. Every vector is a combination of them. So three, four, two can be written three i, plus four j, plus two k. And its magnitude is root of nine plus sixteen plus four. Root twenty-nine. About five point three nine.

The Operations

Componentwise Algebra, Geometric Meaning

Vectors combine through three basic operations. Each has a component definition that makes computation routine. And a geometric interpretation that makes the result intuitive. Addition. Add corresponding components. Subtraction. Subtract corresponding components. And scalar multiplication. Multiply every component by the scalar c. Now the geometry, which is where the intuition lives. Addition places the tail of v at the tip of u. And the sum runs from the tail of u to the tip of v. That's the tip-to-tail rule. Sometimes called the triangle rule. Look at the left panel. u goes out to four comma one. Then v is placed at that tip, going up to five comma four. And the sum, in red, runs directly from the origin to five comma four. Closing the triangle. The middle panel shows the parallelogram rule. Draw both vectors from the origin, complete the parallelogram, and the diagonal is the sum. Same answer. Same vector. Just a different way of seeing it. Scalar multiplication scales the length by absolute c. And reverses the direction when c is negative. And subtraction is just addition of the reversed vector. u minus v equals u plus minus v. The right panel shows that. And notice the dashed arrow — u minus v is precisely the vector pointing from the tip of v, to the tip of u. That's worth remembering. It comes up constantly. Magnitudes scale predictably too. The magnitude of c v equals absolute c, times the magnitude of v.

Three Dimensions

One More Axis, No New Ideas

Everything so far was written for three dimensions. And it specializes to the plane by simply dropping the third component. The three-dimensional coordinate system adds a z-axis, perpendicular to the x y plane. And its orientation is fixed by the right-hand rule. Curl the fingers of your right hand from the positive x-axis toward the positive y-axis. Your thumb points along the positive z-axis. That convention matters, and we'll rely on it again in section ten point three. One nice consequence of the magnitude formula. It is the distance formula. The distance between two points in space is the root of the sum of squared coordinate differences. And that's exactly the magnitude of the vector joining them. Because the vector from P one to P two has the coordinate differences as its components. And its length is the distance between the points. One formula. Two readings. Here's something to think about. A vector is unchanged by translation. Only its components matter. Yet a point is fixed in space — moving it makes it a different point. So why is it useful to have an object that carries direction and magnitude, but deliberately not position? Consider a force of fifty newtons pointing east. It's the same force whether applied at the front of a cart or the back. Even though those are different points.

Key Takeaways

What to Carry Forward

Five takeaways. A vector carries magnitude and direction. Its components are terminal minus initial coordinates. And its magnitude is the root of the sum of squared components. Translation does not change a vector. Only components matter, not where the arrow is drawn. And in standard position, the tip's coordinates equal the components. Dividing by the magnitude gives a unit vector — encoding pure direction, with all size information removed. Operations are componentwise. Add, subtract and scale entry by entry. And the magnitude of c v is absolute c times the magnitude of v. And geometrically, addition is tip-to-tail — equivalently, the parallelogram rule. While u minus v points from the tip of v to the tip of u. Next up is section ten point two — the Dot Product. Which multiplies two vectors into a single number, and measures the angle between them.

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FAQ

Questions students ask us

About Vectors in R2 and R3

How do you find the components and magnitude of a vector between two points?
Subtract the initial point's coordinates from the terminal point's coordinates, then take the square root of the sum of the squared components. For P_1=(x_1,y_1,z_1) and P_2=(x_2,y_2,z_2), the vector is \langle x_2-x_1, y_2-y_1, z_2-z_1\rangle. Its magnitude equals the distance between the points. For example, the vector from (1,1,1) to (4,5,3) is \langle 3,4,2\rangle, with magnitude \sqrt{29}\approx 5.39.
How do you find a unit vector in the direction of a given vector?
Divide the vector by its own magnitude, \mathbf{u}=\mathbf{v}/\|\mathbf{v}\|. The result has length exactly one and points the same way. For \mathbf{v}=\langle 4,3,12\rangle, the magnitude is \sqrt{16+9+144}=\sqrt{169}=13, so the unit vector is \langle 4/13, 3/13, 12/13\rangle. To check, the squared components sum to 169/169=1. Multiplying this unit vector by any positive number produces a vector in the same direction.
What does it mean to write a vector in i, j, k form?
Standard basis form writes a vector as a combination of the unit vectors \mathbf{i}=\langle1,0,0\rangle, \mathbf{j}=\langle0,1,0\rangle, and \mathbf{k}=\langle0,0,1\rangle, which point along the positive x-, y-, and z-axes. The components become the coefficients, so \langle 3,4,2\rangle = 3\mathbf{i}+4\mathbf{j}+2\mathbf{k}. In the plane, you drop the third component and use only \mathbf{i} and \mathbf{j}, so \langle 2,-5\rangle = 2\mathbf{i}-5\mathbf{j}.
Which way does the vector u minus v point?
When both vectors are drawn from the same starting point, \mathbf{u}-\mathbf{v} points from the tip of \mathbf{v} to the tip of \mathbf{u}, not the other way around. This follows because \mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v}), where -\mathbf{v} is \mathbf{v} reversed. Algebraically, you subtract corresponding components, so \langle 4,1\rangle-\langle 1,3\rangle=\langle 3,-2\rangle. Swapping the order gives the opposite vector, since \mathbf{v}-\mathbf{u}=-(\mathbf{u}-\mathbf{v}).

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