Chapter 10

In order to find the equation of a plane, what two pieces of information must one have?

To find an equation of a line, what two pieces of information are needed?

The cross product of two vectors is a _________, not a scalar.

The dot product of two vectors is a ________ not a vector.

Name two different things that cannot be described with just one number, but rather need 2 or more numbers to fully describe them.

Two distinct lines in the plane can intersect or be ________.

How are the concepts of the dot product and vector magnitude related?

What is the difference between (1,2) and \langle 1,2\rangle \(?\)

In Exercises 3-6, give any two points in the given plane. 3\. \(2 x-4 y+7 z=2\)

Two distinct lines in space can intersect, be ________ or be ________.

Give a synonym for "orthogonal."

What is a unit vector?

Give any two points in the given plane. \(3(x+2)+5(y-9)-4 z=0\)

Use your own words to describe what it means for two lines in space to be skew.

T/F: A fundamental principle of the cross product is that \(\vec{u} \times \vec{v}\) is orthogonal to \(\vec{u}\) and \(\vec{v} .\)

Give a synonym for "orthogonal."

Unit vectors can be thought of as conveying what type of information?

In Exercises 5-14, write the vector, parametric and symmetric equations of the lines described. Passes through \(P=(2,-4,1),\) parallel to \(\vec{d}=\langle 9,2,5\rangle\)

_______ is a measure of the turning force applied to an object.

In Exercises 5-10, find the dot product of the given vectors. \(\vec{u}=\langle 2,-4\rangle, \vec{v}=\langle 3,7\rangle\)

What does it mean for two vectors to be parallel?

Consider the hyperbola \(x^{2}-y^{2}=1\) in the plane. If this hyperbola is rotated about the \(x\) -axis, what quadric surface is formed?

Give any two points in the given plane. \(4(y+2)-(z-6)=0\)

Write the vector, parametric and symmetric equations of the lines described. Passes through \(P=(6,1,7),\) parallel to \(\vec{d}=\langle-3,2,5\rangle\).

T/F: If \(\vec{u}\) and \(\vec{v}\) are parallel, then \(\vec{u} \times \vec{v}=\overrightarrow{0} .\)

Find the dot product of the given vectors. \(\vec{u}=\langle 5,3\rangle, \vec{v}=\langle 6,1\rangle\)

What effect does multiplying a vector by -2 have?

Consider the hyperbola \(x^{2}-y^{2}=1\) in the plane. If this hyperbola is rotated about the \(y\) -axis, what quadric surface is formed?

In Exercises 7-20, give the equation of the described plane in standard and general forms. Passes through (2,3,4) and has normal vector \(\vec{n}=\langle 3,-1,7\rangle\)

Write the vector, parametric and symmetric equations of the lines described. Passes through \(P=(2,1,5)\) and \(Q=(7,-2,4)\).

In Exercises 7-16, vectors \(\vec{u}\) and \(\vec{v}\) are given. Compute \(\vec{u} \times \vec{v}\) and show this is orthogonal to both \(\vec{u}\) and \(\vec{v}\). \(\vec{u}=\langle 3,2,-2\rangle, \quad \vec{v}=\langle 0,1,5\rangle\)

Find the dot product of the given vectors. \(\vec{u}=\langle 1,-1,2\rangle, \vec{v}=\langle 2,5,3\rangle\)

In Exercises 7-10, points \(P\) and \(Q\) are given. Write the vector \(\overrightarrow{P Q}\) in component form and using the standard unit vectors. \(P=(2,-1), \quad Q=(3,5)\)

The points \(A=(1,4,2), B=(2,6,3)\) and \(C=(4,3,1)\) form a triangle in space. Find the distances between each pair of points and determine if the triangle is a right triangle.

Give the equation of the described plane in standard and general forms. Passes through (1,3,5) and has normal vector \(\vec{n}=\langle 0,2,4\rangle\)

Write the vector, parametric and symmetric equations of the lines described. Passes through \(P=(1,-2,3)\) and \(Q=(5,5,5)\).

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Compute \(\vec{u} \times \vec{v}\) and show this is orthogonal to both \(\vec{u}\) and \(\vec{v}\). \(\vec{u}=\langle 5,-4,3\rangle, \quad \vec{v}=\langle 2,-5,1\rangle\)

Find the dot product of the given vectors. \(\vec{u}=\langle 3,5,-1\rangle, \vec{v}=\langle 4,-1,7\rangle\)

Points \(P\) and \(Q\) are given. Write the vector \(\overrightarrow{P Q}\) in component form and using the standard unit vectors. \(P=(3,2), \quad Q=(7,-2)\)

The points \(A=(1,1,3), B=(3,2,7), C=(2,0,8)\) and \(D=(0,-1,4)\) form a quadrilateral \(A B C D\) in space. Is this a parallelogram?

Give the equation of the described plane in standard and general forms. Passes through the points (1,2,3),(3,-1,4) and (1,0,1) .

Write the vector, parametric and symmetric equations of the lines described. Passes through \(P=(0,1,2)\) and orthogonal to both \(\vec{d}_{1}=\langle 2,-1,7\rangle\) and \(\vec{d}_{2}=\langle 7,1,3\rangle\)

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Compute \(\vec{u} \times \vec{v}\) and show this is orthogonal to both \(\vec{u}\) and \(\vec{v}\). \(\vec{u}=\langle 4,-5,-5\rangle, \quad \vec{v}=\langle 3,3,4\rangle\)

Points \(P\) and \(Q\) are given. Write the vector \(\overrightarrow{P Q}\) in component form and using the standard unit vectors. \(P=(0,3,-1), \quad Q=(6,2,5)\)

Find the center and radius of the sphere defined by \(x^{2}-8 x+y^{2}+2 y+z^{2}+8=0\)

Give the equation of the described plane in standard and general forms. Passes through the points (5,3,8),(6,4,9) and (3,3,3) .

Write the vector, parametric and symmetric equations of the lines described. Passes through \(P=(5,1,9)\) and orthogonal to both \(\vec{d}_{1}=\langle 1,0,1\rangle\) and \(\vec{d}_{2}=\langle 2,0,3\rangle\)

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Compute \(\vec{u} \times \vec{v}\) and show this is orthogonal to both \(\vec{u}\) and \(\vec{v}\). \(\vec{u}=\langle-4,7,-10\rangle, \quad \vec{v}=\langle 4,4,1\rangle\)

Find the dot product of the given vectors. \(\vec{u}=\langle 1,2,3\rangle, \vec{v}=\langle 0,0,0\rangle\)

Points \(P\) and \(Q\) are given. Write the vector \(\overrightarrow{P Q}\) in component form and using the standard unit vectors. \(P=(2,1,2), \quad Q=(4,3,2)\)