Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes From the Toolbox (what you need from previous classes) Plotting points, sketching vectors. Be able to find the component form a vector given two points. Vector operations: vector addition, scalar multiplication, dot product, and cross product. Know how the dot and cross products are related to orthogonality of vectors. Applications of vector operations: Vector equations of lines in the plane and 3-space (uses vector addition & scalar multiplication). Equations of planes in 3-space (uses the dot & cross products). In this worksheet, you will: Find vector equations r = r 0 + tv and parametric equations for lines in R 2 and R 3. Use the direction vectors of two lines to determine whether or not the lines are parallel. Find the vector equation n P 0 P = 0 of a plane, given a normal vector n and a point P 0 the plane passes through. Use the vector equation to find the algebraic equation ax + by + cz = d. Find an equation of a plane given three points in the plane. Find a normal vector n to a plane given the algebraic equation ax + by + cz = d of the plane. Use the normal vectors of two planes to determine whether or not the planes are parallel.
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes 1 Model 1: Vector & Parametric Equations of Lines Diagram 1A: Slope-Intercept Equation of Line: y = x/3 + 2 Diagram 1B: r 0 = 0, 2, v = 3, 1 Vector Equation of Line: r(t) = r 0 + tv = 0, 2 + t 3, 1 = 3t, t + 2 Diagram 1C: r 0 = 4.5, 0.5, v = 6, 2 Vector Equation of Line: r(t) = r 0 + tv = 4.5, 0.5 + t 6, 2 = 6t 4.5, 2t + 0.5 Critical Thinking Questions In this section, you will compare three types of equations of lines: slope-intercept equations, vector equations, and parametric equations. (Q1) Diagrams 1A, IB, and 1C show three different ways of representing the same line. What is the slope of this line?
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes 2 (Q2) What is the slope of the vector v in Diagram 1B? What is the slope of the vector v in Diagram 1C? (Q3) We will call the vectors v in Diagrams 1B and 1C direction vectors for the line. These vectors are perpendicular / parallel / have no definite relation to the line. (Q4) What are the coordinates (x, y) of the y-intercept of the line in Diagrams 1A, 1B, and 1C? (Q5) In Diagram 1B: The vector r 0 is the position vector of a point P 0 on the line. Find the coordinates of P 0. Then, find the value of t so that r(t) also gives the position of P 0. (Q6) In Diagram 1C: The vector r 0 is the position vector of a point P 0 on the line. Find the coordinates of P 0. Then, find the value of t so that r(t) also gives the position of P 0. (Q7) Compare your answers to (Q5) and (Q6). Complete the following statement: Statement: The vector r(t) is equal to the vector r 0 when t =. (Q8) Compare the vector equations of the line in Diagrams 1B and 1C to the slope-intercept equation in Diagram 1A. In the vector equations, which vector serves a purpose that is similar to the slope in the slope-intercept equation? (Q9) Compare the vector equations of the line in Diagrams 1B and 1C to the slope-intercept equation in Diagram 1A. In the vector equations, which vector serves a purpose that is similar to the y-intercept in the slope-intercept equation? (Q10) Compare the vector equations of the line in Diagrams 1B and 1C to the slope-intercept equation in Diagram 1A. In the slope-intercept equation, the independent variable is x. What is the independent variable in the vector equations? Is this a vector, or a scalar? (This is called the parameter of the vector equation.) (Q11) In Diagram 1B: When t = 0, r(0) = 0, 2. When t = 1, r(1) = 3, 3. Sketch and label these two vectors (based at the origin) on Diagram 1B. The distance traveled along the line between t = 0 and t = 1 equals the magnitude of the vector r 0 / v. (Q12) Diagram 1C: Evaluate the vector function r(t) at t = 0 and t = 1, then sketch these two vectors on Diagram 1C. The distance traveled along the line between t = 0 and t = 1 equals the magnitude of the vector r 0 / v.
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes 3 (Q13) In general, a vector equation of a line is given by: r(t) = r 0 + tv where the vector is parallel to the line, the vector is the position vector for a fixed point on the line, and the scalar is the parameter (variable). The output of the vector equation is the vector points on the line for different values of t.. This vector gives the position of (Q14) These same ideas lead to vector equations of lines in R 3. Find the vector equation for the line in R 3 that passes through the point P = (1, 0, 2) parallel to the vector v = 2, 1, 5. (Q15) The parametric equations (or coordinate functions) of a line are the functions making up the components of the line. For example, the parametric equations in Diagram 1B are: x = 3t, y = t + 2. Write the parametric equations in Diagram 1C: ( Q16) Suppose a line is given by the parametric equations x = 1 + 3t and y = 1 2t. (a) Write the vector equation of this line, in the form r(t) = r 0 + tv. (b) On the axes below, sketch this line and the vectors v and r 0. (c) At what t-value does this line cross the y-axis? What is the y-coordinate where this line crosses the y-axis?
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes 4 ( Q17) Write a vector equation and parametric equations for this line. Vector equation: Parametric equations: ( Q18) List at least two things that you can do with vector equations of lines, that you cannot do with a slope-intercept equation. Model 2: Parallel Lines & Direction Vectors DIAGRAM 2: L 1 and L 2 are parallel lines. The direction vector of L 1 is v = 4, 2 The direction vector of L 2 is w = 2, 1 Critical Thinking Questions In this section, you will use direction vectors to determine whether or not two lines are parallel. (Q19) The direction vectors v and w in Diagram 2 are: perpendicular / parallel / neither. (Q20) Recall that two vectors are parallel if (and only if) they are scalar multiples of each other. Since the direction vectors v and w in Diagram 2 are parallel, there is a scalar c such that v = cw. Find c. (Q21) Suppose you have two lines with vector equations, r 1 (t) = r 01 + tv 1 and r 2 (t) = r 02 + tv 2. Using these equations, how can you tell whether or not the two lines are parallel?
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes 5 ( Q22) Which of these lines are parallel to the lines L 1 and L 2 in Diagram 2? (a) r(t) = 1, 1 + t 6, 3 parallel / not parallel (b) r(t) = 2, 1 + t 5, 7 parallel / not parallel (c) r(t) = 108 2t, 33/7 t parallel / not parallel (d) r(t) = 0, 2 + t 1, 2 parallel / not parallel Model 3: Normal Vectors & Equations of Planes in R 3 Algebraic Equation of Plane: Diagram 3A: 3x + 4y + 6z = 12 Vector Equation of Plane: 3, 4, 6 x 2, y 0, z 1 = 0 n = 3, 4, 6 : normal vector P 0 = (2, 0, 1): fixed point on plane P = (x, y, z): arbitrary point on plane P 0 P = x 2, y 0, z 1 : vector in plane Algebraic Equation of Plane: Diagram 3B: ax + by + cz = d Vector Equation of Plane: n P 0 P = 0 n = a, b, c : normal vector P 0 = (x 0, y 0, z 0 ): fixed point on plane P = (x, y, z): arbitrary point on plane P 0 P = x x 0, y y 0, z z 0 : vector in plane (The textbook uses r r 0 instead of P 0 P. Different notation, but the same vector.)
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes 6 Critical Thinking Questions In this section, you will use a normal vector and a fixed point in a plane to derive the algebraic equation of a plane. (Q23) Diagram 3A shows the part of the plane 3x + 4y + 6z = 12 that passes through the first octant of R 3 (this is where all three coordinate axes are positive). The vector n = 3, 4, 6 is a normal vector to this plane; it is orthogonal to every vector in the plane. The points P = (x, y, z) and P 0 = (2, 0, 1) lie in this plane, so the vector P 0 P = x 2, y, z 1 lies in the plane. Sketch the vector P 0 P on Diagram 3A. (Q24) Since n = 3, 4, 6 is normal to the plane, and P 0 P = x 2, y, z 1 lies in the plane, then the dot product of n and P 0 P must equal. (Q25) Computing the dot product in the equation n P 0 P = 0 and re-writing the equation, you can see that the vector and algebraic equations for the plane in Diagram 3A are equivalent: n P 0 P = 0 3, 4, 6 x 2, y, z 1 = 0 3x 6 + y + 6 = 0 x + y + z = (Q26) Diagram 3B shows a generic plane, with normal vector n = a, b, c, a fixed point P 0 = (x 0, y 0, z 0 ) in the plane, and an arbitrary point P = (x, y, z) in the plane. The vector P 0 P = x x0, y y 0, z z 0 lies in the plane. Since n = a, b, c is normal to the plane, and P 0 P = x x 0, y y 0, z z 0 lies in the plane, then n P 0 P =. (Q27) Computing the dot product in the equation n P 0 P = 0 and re-writing the equation, you can see that the vector and algebraic equations for the plane in Diagram 3B are equivalent:, b, n P 0 P = 0 x x 0,, = 0 (x x 0 ) + b (y y 0 ) + ( ) = 0 x + b + = x 0 + y 0 + z 0 x + y + z = d where d = ax 0 + by 0 + cz 0
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes 7 For a plane with normal vector n = a, b, c, and containing the point P 0 = (x 0, y 0, z 0 ), the vector and algebraic equations are equivalent: Vector Equation: n P 0 P = 0 or: a, b, c x x 0, y y 0, z z 0 = 0 Algebraic Equation: ax + by + cz = d where: d = ax 0 + by 0 + cz 0 (Q28) Suppose P is a plane passing through the point P 0 = (3, 1, 9), and n = 1, 5, 2 is a normal vector to P. Write the vector equation of this plane. (Remember to use P 0 = (3, 1, 9) and P = (x, y, z) to find P 0 P.) (Q29) Use your vector equation from (Q28) to find the algebraic equation of the plane P. (Q30) An Important Observation: The components of the normal vector n = a, b, c are the same as / different than the coefficients of x, y, and z in the algebraic equation of the plane. (Q31) Let P be the plane with equation 4x 2y + 7z = 42. Find a normal vector n for this plane. ( Q32) Two planes are parallel if they have the same normal vectors. Which of the following equations are planes parallel to the plane P from (Q31)? (circle all correct answers:) 4x 2y + 7z = 107 2x 7y + 4z = 42 8x + 4y 14z = 0 ( Q33) The equations 8x 4y + 14z = 42 and 4x 2y + 7z = 42 describe planes that are: not parallel / parallel but distinct (not the same plane) / the same plane. Explain how you decided on your answer. ( Q34) A normal line to a plane is the line that is perpendicular to the plane and passes through the origin O = (0, 0, 0). Find a vector equation for the normal line to the plane in (Q31).
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes 8 ( Q35) Suppose you have a plane that contains three points: P 0, Q 0, and R 0, as pictured on the right. Sketch the two vectors P 0 Q 0 and P 0 R 0. Using the vectors P 0 Q 0 and P 0 R 0, which one of the following vector operation can you use to find a normal vector to this plane? Vector Addition / Scalar Multiplication / Dot Product / Cross Product ( Q36) Suppose P is a plane containing the points P 0 = (3, 0, 0), Q 0 = (0, 1, 0), and R 0 = (0, 0, 4). Use your answer from (Q35) to find a normal vector to the plane. ( Q37) Now, find the algebraic equation for the plane in (Q36).
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes 9 Summary To find the vector equation of a line, you need to know: A vector v that is A P 0 that is on the line. to the line. In the vector equation of a line r = r 0 + tv, the vector v gives the of the line, P 0 is a on the line, and r 0 is the position vector of the point. Parallel lines have direction vectors. So if r 1 = r 01 +tv 1 and r 2 = r 02 +tv 2 are parallel, then v 1 and v 2 are multiples of each other. A normal vector to a plane is a vector that is to the plane. To find the vector equation of a plane, you need to know: A vector n that is A to the plane. P 0 that is on the plane. If a plane has the algebraic equation ax + by + cz = d, then a normal vector to the plane is n =. Parallel planes have normal vectors. So if n 1 is normal to the plane P 1, and n 2 is normal to the plane P 2, then n 1 and n 2 are multiples of each other. If you know three points that are on a plane, you can find a normal vector to the plane by making vectors, then using the.