3-D Cartesian Coordinate System YASIR ALI VECTOR CALCULUS 3-D Cartesian Coordinate System YASIR ALI VECTOR CALCULUS 3-D Cartesian Coordinate System YASIR ALI VECTOR CALCULUS VECTORS YASIR ALI VECTOR CALCULUS VECTORS YASIR ALI VECTOR CALCULUS Vectors A vector a in 3-dimensional space may be represented as a = ax^i+ ay^j + az ^ k = [ax , ay , az ]. We will use a1 , a2 and a3 instead of ax , ay and az , respectively. Magnitude of a: |a| = q a12 + a2 2 + a3 2 . YASIR ALI VECTOR CALCULUS Position Vectors The four arrows in the plane (directed line segments) shown here have the same length and direction. They therefore represent the same vector, YASIR ALI VECTOR CALCULUS Position Vectors A vector PQ in standard position has its initial point at the origin. The directed line segments PQ and ~v are parallel and have the same length. YASIR ALI VECTOR CALCULUS Vector Addition Let u = [u1 , u2 , u3 ] and v = [v1 , v2 , v3 ] be vectors Addition: u + v = [u1 + v1 , u2 + v2 , u3 + v3 ] YASIR ALI VECTOR CALCULUS Scalar Multiplication YASIR ALI VECTOR CALCULUS In component form, the vector from P1 (x1 , y1 , z1 ) to P2 (x2 , y2 , z2 ) is P1 P2 = (x2 − x1 )î + (y2 − y1 )ĵ + (z2 − z1 )k̂ P1 P2 = [(x2 − x1 ), (y2 − y1 ), (z2 − z1 )] YASIR ALI VECTOR CALCULUS In component form, the vector from P1 (x1 , y1 , z1 ) to P2 (x2 , y2 , z2 ) is P1 P2 = (x2 − x1 )î + (y2 − y1 )ĵ + (z2 − z1 )k̂ P1 P2 = [(x2 − x1 ), (y2 − y1 ), (z2 − z1 )] Unit Vector v̂ = YASIR ALI v |v| VECTOR CALCULUS In component form, the vector from P1 (x1 , y1 , z1 ) to P2 (x2 , y2 , z2 ) is P1 P2 = (x2 − x1 )î + (y2 − y1 )ĵ + (z2 − z1 )k̂ P1 P2 = [(x2 − x1 ), (y2 − y1 ), (z2 − z1 )] Unit Vector v̂ = v |v| Find a unit vector u in the direction of the vector from P1 (1, 0, 1) to P2 (3, 2, 0) 1 Find P1 P2 2 Find |P1 P2 | 3 Find û = u |u| YASIR ALI VECTOR CALCULUS Dot Product a.b = |a||b| cos θ, where θ represents the shortest angle between a and b. If a and b are given then we can find angle between them by cos θ = a.b . |a||b| Let a and b be given as follows a = a1^i+ a2^j + a3^ k ^ ^ b = b1 i + b2 j + b3^ k then dot product of a and b is a.b = a1 b1 + a2 b2 + a3 b3 YASIR ALI VECTOR CALCULUS A small cart is being pulled along a smooth horizontal floor with a 20-lb force F making a 450 angle to the floor. What is the effective force moving the cart forward? YASIR ALI VECTOR CALCULUS Find the angle θ in the triangle ABC determined by the vertices A = (0, 0), B = (3, 5), and C = (5, 2) YASIR ALI VECTOR CALCULUS CROSS PRODUCT a × b = |a||b| sin θ.^ n, where n̂ is perpendicular to plane of a and b. In component form the cross product of a and b is given by a×b= ^i ^j ^ k a1 a2 a3 b1 b2 b3 or a×b= (a2 b3 − b2 a3 )^i− (a3 b1 − b1 a3 )^j− (a1 b2 − b1 a2 )^ k. YASIR ALI VECTOR CALCULUS CROSS PRODUCT YASIR ALI VECTOR CALCULUS Find a vector perpendicular to the plane of P(1, −1, 0), Q(2, 1, −1), and R(−1, 1, 2) YASIR ALI VECTOR CALCULUS Area of Parallelogram |u × v| represents the area of a parallelogram with sides |u| and |v| YASIR ALI VECTOR CALCULUS Lines Parametric equations of straight line passing through point (x0 , y0 , z0 ) and parallel to a = [a1 , a2 , a3 ] are x = x 0 + a1 t y = y 0 + a2 t z = z 0 + a3 t (t ∈ R). In vector form r(t) = r0 + vt. YASIR ALI VECTOR CALCULUS Find the line that passes through the point P1 (x1 , y1 , z1 ) and that is parallel to the position vector a = [a1 , a2 , a3 ] P1 P are parallel to OA P1 P = tOA YASIR ALI VECTOR CALCULUS Find the line that passes through the point P1 (x1 , y1 , z1 ) and that is parallel to the position vector a = [a1 , a2 , a3 ] P1 P are parallel to OA P1 P = tOA [x − x1 , y − y1 , z − z1 ] = t[a1 , a2 , a3 ] x = x1 + a1 t y = y1 + a2 t z = z1 + a3 t YASIR ALI VECTOR CALCULUS Find parametric equations for the line through P(−2, 0, 4) and parallel to v = 2î + 4ĵ − 2k̂. YASIR ALI VECTOR CALCULUS Find parametric equations for the line through P(−2, 0, 4) and parallel to v = 2î + 4ĵ − 2k̂. Requirements 1 Base Point 2 Direction given P given v YASIR ALI VECTOR CALCULUS Find parametric equations for the line through P(−2, 0, 4) and parallel to v = 2î + 4ĵ − 2k̂. Requirements 1 Base Point 2 Direction given P given v P(−2, 0, 4) & v = [2, 4, −2] x = −2 + 2t y = 0 + 4t z = 4 − 2t Find parametric equations for the line through P(−2, 0, 4) and parallel to v = 2î + 4ĵ − 2k̂. Requirements 1 Base Point 2 Direction given P given v P(−2, 0, 4) & v = [2, 4, −2] x = −2 + 2t z = 4 − 2t Find parametric equations for the line through P(−2, 0, 4) and parallel to v = 2î + 4ĵ − 2k̂. Requirements 1 Base Point 2 Direction given P given v P(−2, 0, 4) & v = [2, 4, −2] x = −2 + 2t y = 0 + 4t z = 4 − 2t YASIR ALI VECTOR CALCULUS Find parametric equations for the line through P(−3, 2, −3) and Q(1, −1, 4). YASIR ALI VECTOR CALCULUS Find parametric equations for the line through P(−3, 2, −3) and Q(1, −1, 4). Requirements YASIR ALI VECTOR CALCULUS Find parametric equations for the line through P(−3, 2, −3) and Q(1, −1, 4). Requirements 1 Base Point YASIR ALI VECTOR CALCULUS Find parametric equations for the line through P(−3, 2, −3) and Q(1, −1, 4). Requirements 1 Base Point 2 Direction any of P or Q may be selected as base point YASIR ALI VECTOR CALCULUS Find parametric equations for the line through P(−3, 2, −3) and Q(1, −1, 4). Requirements 1 Base Point 2 Direction any of P or Q may be selected as base point PQ YASIR ALI VECTOR CALCULUS Find parametric equations for the line through P(−3, 2, −3) and Q(1, −1, 4). Requirements 1 Base Point 2 Direction any of P or Q may be selected as base point PQ Q(1, −1, 4) & PQ = [4, −3, 7] YASIR ALI VECTOR CALCULUS Find parametric equations for the line through P(−3, 2, −3) and Q(1, −1, 4). Requirements 1 Base Point 2 Direction any of P or Q may be selected as base point PQ Q(1, −1, 4) & PQ = [4, −3, 7] x = 1 + 4t y = −1 − 3t z = 4 + 7t YASIR ALI VECTOR CALCULUS Parametric Equations of Line Can you find parametric equations of line for through two given points: P(1, 2, −1) and Q(−1, 0, 1). A ~ = [−2, −2, 2] vector parallel to required line is PQ parametric eq. x = 1 − 2t, y = 2 − 2t and z = −1 + 2t through a point and a given parallel vector:(or parallel line) P(1, 2, −1) and [0, 2, 1] then parametric eq. x = 1, y = 2 + 2t and z = −1 + t through a point and parallel to given line: Q(−1, 0, 1) and parallel line x = 1 − 2t, y = 2 − 2t and z = −1 + 2t. A vector parallel to required line is [−2, −2, 2] parametric eq. x = −1 − 2t, y = −2t and z = 1 + 2t through a point and parallel to a given axis: P(1, 2, −1) and parallel to x−axis. A vector parallel to required line is [1, 0, 0] parametric eq. x = 1 + t, y = 2 and z = −1 YASIR ALI VECTOR CALCULUS PLANES If n = [a1 , a2 , a3 ] to a plane and P0 (x0 , y0 , z0 ) is any point on the plane then equation of plane is −−→ n.P0 P = 0 a1 (x − x0 ) + a2 (y − y0 ) + a3 (z − z0 ) = 0, where P(x, y , z) is any point on the plane. In general a1 x + a2 y + a3 z + d = 0. YASIR ALI VECTOR CALCULUS Find an equation of the plane containing the point (1, 2, 3) with normal vector [4, 5, 6], and sketch the plane. 1 Point on Plane 2 Normal to the Plane 3 Use a1 (x − x0 ) + a2 (y − y0 ) + a3 (z − z0 ) = 0 YASIR ALI VECTOR CALCULUS Find an equation of the plane containing the point (1, 2, 3) with normal vector [4, 5, 6], and sketch the plane. 1 Point on Plane 2 Normal to the Plane 3 Use a1 (x − x0 ) + a2 (y − y0 ) + a3 (z − z0 ) = 0 Find the plane containing the three points A(1, 2, 2), B(2, −1, 4) and C (3, 5, −2). 1 Point on Plane 2 Normal to the Plane 3 Use a1 (x − x0 ) + a2 (y − y0 ) + a3 (z − z0 ) = 0 YASIR ALI VECTOR CALCULUS Find an equation of the plane containing the point (1, 2, 3) with normal vector [4, 5, 6], and sketch the plane. 1 Point on Plane 2 Normal to the Plane 3 Use a1 (x − x0 ) + a2 (y − y0 ) + a3 (z − z0 ) = 0 Find the plane containing the three points A(1, 2, 2), B(2, −1, 4) and C (3, 5, −2). 1 Point on Plane 2 Normal to the Plane 3 Use a1 (x − x0 ) + a2 (y − y0 ) + a3 (z − z0 ) = 0 Find AB and AC Find AB × AC YASIR ALI VECTOR CALCULUS NON-COLLINEAR POINTS AB = [1, −3, 2] and AC = [1, 6, −6] î ĵ k̂ AB × AC = 1 −3 2 1 6 −6 YASIR ALI VECTOR CALCULUS NON-COLLINEAR POINTS AB = [1, −3, 2] and AC = [1, 6, −6] î ĵ k̂ AB × AC = 1 −3 2 1 6 −6 n̂ = [6, 8, 9] P(1, 2, 2) and n̂ = [6, 8, 9] 6(x − 1) + 8(y − 2) + 9(z − 2) = 0 YASIR ALI VECTOR CALCULUS Angle Between Two planes YASIR ALI VECTOR CALCULUS INTERSECTING PLANES 1 Parallel planes have same normal vector. 2 Intersection of the planes forms a line. YASIR ALI VECTOR CALCULUS Find the line of intersection of the planes x + 2y + z = 3 and x − 4y + 3z = 5. 1 Solving both equations for a single variable (say x) x = 3 − 2y − z, 2 & x = 5 + 4y − 3z Equating both expression gives value of one of the remaining variables 3 − 2y − z = 5 + 4y − 3z =⇒ z = 3y + 1 3 Using this value in the first expression gives x = −5y + 2. 4 Now letting y = t gives x = −5t + 2, y = t, YASIR ALI and z = 3t + 1 VECTOR CALCULUS Find a vector parallel to the line of intersection of the planes 3x − 6y − 2z = 15 and 2x + y − 2z = 5. YASIR ALI VECTOR CALCULUS Distance from a point to a line and a plane Distance from a point to a line: The distance from a point S to a line passing through a point P and parallel to a is |PS × a| . d= |a| Distance from a point to a plane: Distance from a point P0 (x0 , y0 , z0 ) to the plane ax + by + cz + d = 0 is d= |ax0 + by0 + cz0 + d| √ . a2 + b 2 + c 2 YASIR ALI VECTOR CALCULUS