MTH 254 Exam 1

Scalar

A physical quantity that has magnitude only, no direction (a real number)

Vector

A quantity that has magnitude and direction

Magnitude

Greatness of size, length

What is the formula to find a vector of magnitude k in the same direction as a vector? Find a vector in the same direction as v = <3, 2, 3> of magnitude 5

(k / ||v||) * v ||v|| = √3^2 + 2^2 + 3^2 = √22 k = 5 (5 /√22) * <3, 2, 3> = <15/√22, 10/√22, 15/√22>

What is the equation for a sphere with a center (a, b, c) and radius r?

(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2

What is the formula for the area of the triangle?

0.5 * ||v x w||

What is the formula to find a unit vector in the same direction as a vector? Find a unit vector in the same direction as v = <-4, 2, 1>

1 / ||<v1, v2, v3>|| * <v1, v2, v3> ||√(-4)^2 + 2^2 + 1^2|| = √16 + 4 + 1 = √21 1 / √21 * <-4, 2, 1> = <-4/√21, 2/√21, 1/√21>

Find the intersection between these two lines: 𝑙1: r1 = (5 + 2t1)i + (43−7t1)j + 2t1k 𝑙2: r2=(3 + 2t2)i + (15 − 2t2)j + (19−t2)k

A vector equation of a line in the direction D ≠ 0 and containing the point P0=(𝑥0,𝑦0,𝑧0) is r = r0 + tD, for some scalar t, where r0 is a vector OP, P = (x, y, z) is any point on the line, and O is the origin of the coordinate system. To find the point of intersection of 𝑙1 and 𝑙2, determine the values of t1 and t2. Equate the components of each vector to obtain the following simultaneous system 5 + 2t1 = 3 + 2t2 43 - 7t1 = 15 - 2t2 2t1 = 19 - t2 Therefore, from the third equation t2 = 19 − 2t1. Substitute this into the first equation to find t1. 5 + 2t1 = 3 + 2(19 - 2t1) 5 + 2t1 = 3 + 38 - 4t1 6t1 = 36 t1 = 6 From the third equation now you can find t2. t2 = 19 - 2 * 6 t2 = 7 To be clear that both lines intersect, check whether the values of t1 and t2 satisfy the second equation. 43 - 7t1 = 15 - 2t2 43 - 6 ⋅ 7 = 15 - 2 ⋅ 7 1 = 1 Determine the point of intersection by substituting t2 = 7 into the equation for l2, or, alternatively, by substituting t1=6 into the equation for l1. (3 + 2 ⋅ 7)i + (15 - 2 ⋅ 7)j + (19 - 7)k So, the intersection point is (17, 1, 12).

What is the formula for finding a vector with two points? Find component form of a vector with initial point (3, 8, 2) and terminal point (2, -1, 3)

End - initial = (x2 - x1, y2 - y1) (2 - 3, -1 - 8, 3 - 2) = <-1, -9, 1>

How can you accurately find the rays which form the shadow of the vector?

Find the lines perpendicular to the vector being projected onto. This can be done by taking the inverse of a given slope. E.g. if a slope is 3, the slope perpendicular to it will be -1/3

Find the point P of intersection of the plane x + y − z = 5 and line (x + 3)/2 = (y - 4) / 1 = z / 2:

First, convert the symmetric equations for the line to parametric form. The direction vector of the line is 2i + j + 2k and the line passes through the point (−3,4,0). Therefore, the parametric equations are x = −3 + 2𝑡 y = 4 + 𝑡 z = 2𝑡 To find the point on the line that also lies in the plane, substitute the parametric equations for x, y, and z into the equation of the plane and solve for t. x + y - z = 5 (-3 + 2t) + (4 + t) - 2t = 5 t + 1 = 5 so that t = 4. It follows that the line intersects the plane when 𝑡 = 4 in the parametric equations. Thus, the required point is P = (x, y, z) = (−3 + 2 * 4, 4 + 4, 2 * 4) = (5, 8, 8)

When are two vectors parallel?

If they are scalar multiples of each other. That is, (x1, y1) and (x2, y2) are parallel if and only if there is a number k such that (x1, y1) = k(x2, y2).

What is the formula for the addition of two vectors connected head to tail (algebraically & geometrically)?

In algebra: <v1, v2, v3> + <u1, u2, u3> = <v1 + u1, v2 + u2, v3 + u3> In geometry: The new vector will form a triangle, from the tail of the two vectors to their head

What is the formula for the subtraction of two vectors connected head to tail (algebraically & geometrically)?

In algebra: <v1, v2, v3> - <u1, u2, u3> = <v1 - u1, v2 - u2, v3 - u3> In geometry: Reverse the direction of the vector being subtracted, placing its tail on the end of the head of the other vector. The combination of those two vectors will be the vector that forms a triangle from the tail of those two vectors to their head.

What is the formula for the addition of two vectors connected tail to tail (algebraically & geometrically)?

In algebra: ||<v1 + u1, v2 + u2, v3 + u3>|| In geometry: Create a parallelogram by mirroring each vector onto the head of the other, those two vectors should meet at a point. Draw a line from the tails of the original vectors to the heads of the new vectors, that is the vector from their addition.

What is the formula for the magnitude of the cross product with an angle?

Let u and v be vectors, and let 𝜃 be the angle between them. Then, ‖u×v‖ = ‖u‖ ⋅ ‖v‖ ⋅ sin𝜃

Find the intersection between two planes: x + 2y + z = 5 2x + y - z = 7

Set z = 0 or x = 0 or y = 0: x + 2y = 5 2x + y = 7 Multiply by -2 then solve using system of equations: -2(x + 2y) = (5)-2 2x + y = 7 -2x - 4y = -10 2x + y = 7 -3y = -3 y = 1 2x + y = 7 2x + 1 = 7 2x = 6 x = 3 So P(3, 1, 0) You can find another point by doing the same thing but set x = 0.

How do you find a vector normal to a plane that contains two vectors? The following lines intersect at a single point p. l1 => x = t, y = 2t + 1, z = 3t + 4 l2 => x = 2s - 2, y = 2s - 1, z = 3s + 1 Find the coordinates of P, and then find an equation of the plane P that contains both lines.

Use the cross product, the result of those two vectors is a vector normal to the plane. t = 2s - 2 2t + 1 = 2s - 1 => 2(2s - 2) + 1 = 2s - 1 => 4s - 4 + 1 = 2s - 1 => 4s - 3 = 2s - 1 => 2s = 2 => s = 1 t = 2(1) - 2 => t = 0 x = 0, y = 2(0) + 1 => y = 1, z = 3(0) + 4 => z = 4 (0, 1, 4) <1, 2, 3> x <2, 2, 3> (6 - 6) - (3 - 6) + (2 - 4) => <0, 3, -2> ax + by + cz = d 0x + 3y - 2z = d 0(0) + 3(1) - 2(4) = d 3 - 8 = d -5 3y - 2z = -5

What is the formula for the equation of a plane who passes through the point of r0 = (x0, y0, z0) and has the normal vector of n = <a, b, c>? Vector form: Scalar form: General form: Find the scalar equation and general form from the points P = (3, -1, 2), Q = (4, -2, 1), R = (-5, 3, -1):

Vector form: <a, b, c> * <x - x0, y - y0, z - z0> = 0 Scalar equation: a(x - x0) + b(y - y0) + c(z - z0) = 0 General form: ax + by + cz + d = 0 PQ = <4 - 3, -2 + 1, 1 - 2> = <1, -1, -1> PR = <-5 - 3, 3 + 1, -1 - 2> = <-8, 4, -3> Cross product: <3 + 4, 8 + 3, 4 - 8> Normal vector = <7, 11, -4> Use any point as a vector and the normal vector to construct the scalar equation (lets use point P, (3, -1, 2) becomes <3, -1, 2>): a(x - x0) + b(y - y0) + c(z - z0) = 0 7(x - 3) + 11(y - 1) + -4(z - 2) = 0 General equation: 7x - 21 + 11y + 11 - 4z + 8 = 0 7x + 11y - 4z - 2 = 0

What is the formula for the equation of the line where r0 = (x0, y0, z0) is a point on the line and v = <a, b, c> is a vector in the direction of the line: Vector form: Parametric form: Symmetric form: Find the equation of the line through the initial point of (1, -3, 2) and terminal point of (5, -2, 8):

Vector form: <x, y, z> = (x0, y0, z0) + t<a, b, c>' Parametric form: x = x0 + t * a y = y0 + t * b z = z0 + t * c Symmetric Form (x - x0) / a = (y - y0) / b = (z - z0) / c Parallel vector (vector that connects the two points together): <5 - 1, -2 + 3, 8 - 2> = <4, 1, 6> Use either point for the r0 vector: r0 = <1, -3, 2> Vector equation: <x, y, z> = <1, -3, 2> + <4, 1, 6> * t Parametric equations: x = 1 + 4t, y = -3 + t, z = 2 + 6t Symmetric equations: (x - 1) / 4 = (y + 3) = (z - 2) / 6

Find if these vectors intersect? F(t) = <2 + 2t, 8 + t, 10 + 3t> R(t) = <6 + s, 10 - 2s, 16 - s>

We get: 2 + 2t = 6 + s 8 + t = 10 - 2s 10 + 3t = 16 - s Work system of equations with any two: 2 + 2t = 6 + s => 2t = 4 + s => t = 2 + 5/2 8 + t = 10 - 2s => 8 + 2 + 5/2 = 10 - 2s => 10 + 5/2 = 10 - 2s => 5/2 = -2s => s = -4s => 0 = -3s = s = 0 t = 2 + 0/2 => t = 2 Does t = 2 solve all three? Yes! This means there is an intersection point, if no, then there would be no intersection. F(t) = <2 + 2t, 8 + t, 10 + 3t> => <6, 10, 16> R(t) = <6 + s, 10 - 2s, 16 - s> => <6, 10, 16> Thus our point of intersection is (6, 10, 16)

When are two vectors orthogonal? Find x so that vector p = <2, 8, -1> and vector q = <x, -1, 2> are orthogonal:

When their dot product is 0. p * q = 0 => p * q = 2 * x + 8 * -1 + -1 * 2 = 2x - 8 - 2 2x - 8 - 2 = 0 => 2x = 10 => x = 5

What is the formula for multiplying a vector by a scalar?

a * <v1, v2, v3> = <a * v1, a * v2, a * v3>

What is the formula for the angle between two vectors? Find the angle between two vectors: a = <1, 2, 0>, b = < 2, 4, 1>

cos𝜃 = (v⋅w) / (||v|| * ||w||) a * b = 1 * 2 + 2 * 4 + 0 * 1 = 2 + 8 + 0 = 10 || a || = √1^2 + 2^2 + 0^2 = √1 + 4 + 0 = √5 || b || = √2^2 + 4^2 + 1^2 = √4 + 16 + 1 = √21 cos Θ = (u * v) / (||u|| * ||v||) cos Θ = 10 / (√5 * √21) => Θ = cos-1(10/(√5 * √21)) => Θ = 0.22 rad

What is the formula for the distance from a point P = (x0, y0, z0) to a plane ax + by + cz + d = 0? Find the distance d from the point (5,1,−1) to the plane 2x − y + z = 1:

d = |ax0 + by0 + cz0 + d| / √a^2 + b^2 + c^2 d = |2 ⋅ 5 − 1 ⋅ 1 + 1 ⋅ (−1) − 1| / √2^2 + (-1)^2 + 1^2 = 7/√6

Let L be a line in space passing through point P with direction vector v. If M is any point not on L, then what formula describes the distance from M to L (distance from a point to a line)?

d = ||PM x v|| / ||v||

What is the formula for the distance between two points? Find the distance between (1, -5, 4) and (4, -1, -1).

d = √ (x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2 (4, -1, -1) - (1, -5, 4) = <4 - 1, -1 - (-5), -1 - 4> = <3, 4, -5> √3^2 + 4^2 + (-5)^2 = √9 + 16 + 25 = √50

What are the standard unit vectors?

i = <1, 0, 0>, j = <0, 1, 0>, k = <0, 0, 1>

What is the formula for the vector projection of vector u onto vector v? Find the projection of u onto v where u = <5, -3, 1> and v = <2, 3, 2>

proj v U = ((v * u) / (u * u)) * u proj v U = (<5, -3, 1> * <2, 3, 2>) / (<2, 3, 2> * <2, 3, 2>) * <2, 3, 2> => (5(2) + (-3(3) + 2(1)) / (2(2) + 3(3) + 2(2)) * <2, 3, 2> => (10 - 9 + 2) / (4 + 9 + 4) * <2, 3, 2> => (3 / 17) * <2, 3, 2> => <6 / 17, 9 / 17, 6 / 17>

What is the formula for the triple scalar product of between 3 vectors?

u * (v x w)

What is the formula for the cross product? Find the cross product of <5, 1, 2> and <-2, 3, 1>

u x w = (v2u3 - u2v3) - (v1u3 - u1v3) + (v1u2 - u1v2) <1 * 1 - 3 * 2, -2 * 2 - 5 * 1, 5 * 3 - (-2) * 1> = <1 - 6, -4 - 5, 15 + 2> = <-5, -9, 17>

What is the formula for the dot product? Find the dot product between u = <2, 9, -1>, v = <-3, 1, -4>.

v1u1 + v2u2 + v3u3 u * v = <2 * -3 + 9 * 1 + -1 * -4> = <-6 + 9 + 4> = 7

What is the equation for xy, xz, yz plane for point (a, b, c)?

xy-plane: z = c xz-plane: y = b yz-plane: x = a

What is the formula for magnitude? Find magnitude of v = <-1, 4, 3>

||<v1, v2, v3>|| => √(v1)^2 + (v2)^2 + (v3)^2 ||<-1, 4, 3>|| = √(-1)^2 + 4^2 + 3^2 = √1 + 16 + 9 = √26

What is the formula for the area of the parallelogram?

||v x w||

Find the distance d between the two parallel planes x + 2y − 2z = -5 and x + 2y - 2z = 7:

π1: ax + by + cz + d1 = 0 π2: ax + by + cz + d2 = 0 |d2 - d1|/√(a^2 + b^2 + c^2) d = |5 + 7| / √1^2 + 2^2 + (-2)^2 = 12/3 = 4

What is the formula for the volume of a parallelepiped (3D parallelogram)?

𝑉 = |u * (v x w)|