?If v1, v2, and v3 are noncoplanar vectors, let (These vectors occur in the study of

Find the cross product \(a \times b\) and verify that it is orthogonal to both a and b. \(a=\langle 2,3,0\rangle, b=\langle 1,0,5\rangle\) Equation Transcription: ???? Text Transcription: a times b a=langle 2, 3, 0 rangle, b=langle 1, 0, 5 rangle

Find the cross product and verify that it is orthogonal to both and .

Find the cross product and verify that it is orthogonal to both and .

Find the cross product and verify that it is orthogonal to both and .

Find the cross product \(a \times b\) and verify that it is orthogonal to both a and b. \(a=\frac{1}{2} i+\frac{1}{3} j+\frac{1}{4} k, \ \ b=i+2 j-3 k\)

Find the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that it is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). \(\mathbf{a}=t \mathbf{i}+\cos t \mathbf{j}+\sin t \mathbf{k}, \quad \mathbf{b}=\mathbf{i}-\sin t \mathbf{j}+\cos t \mathbf{k}\) Equation Transcription: ???? ???? ???? ???? ???? ???? ???? ????. ???????? ???? ???? Text Transcription: a x b a b a=ti+cos tj+sin tk,b=i-sin tj+cos tk

Find the cross product and verify that it is orthogonal to both and .

If \(a=i-2 k\) and \(b=j+k\), find \(a \times b\). Sketch a, b and \(a \times b\) as vectors starting at the origin.

Find the vector, not with determinants, but by using properties of cross products.

Find the vector, not with determinants, but by using properties of cross products. \(\mathbf{k} \times(\mathbf{i}-2 \mathbf{j})\)

Find the vector, not with determinants, but by using properties of cross products.

Find the vector, not with determinants, but by using properties of cross products.

State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. (a) (b) (c) (d) (e) (f)

Find \(|u \times v|\) and determine whether \(u \times v\) is directed into the page or out of the page.

Find \(|u \times v|\) and determine whether \(u \times v\) is directed into the page or out of the page.

The figure shows a vector a in the -plane and a vector in the direction of . Their lengths are and (a) Find . (b) Use the right-hand rule to decide whether the components of are positive, negative, or 0 .

If and , find and .

If , and , show that .

Find two unit vectors orthogonal to both and .

Find two unit vectors orthogonal to both and .

Show that for any vector a in .

Show that for all vectors and in .

Prove the specified property of cross products (Theorem 11). Property a

Prove the specified property of cross products (Theorem 11). Property

Prove the specified property of cross products (Theorem 11). Property 3:

Prove the specified property of cross products (Theorem 11). Property 4:

Find the area of the parallelogram with vertices , , and .

Find the area of the parallelogram with vertices P(1, 0, 2), Q(3, 3, 3), R(7, 5, 8), and S(5, 2, 7).

(a) Find a nonzero vector orthogonal to the plane through the points , and , and (b) find the area of triangle .

(a) Find a nonzero vector orthogonal to the plane through the points , and , and (b) find the area of triangle .

(a) Find a nonzero vector orthogonal to the plane through the points , and , and (b) find the area of triangle .

(a) Find a nonzero vector orthogonal to the plane through the points , and , and (b) find the area of triangle .

Find the volume of the parallelepiped determined by the vectors , and .

Find the volume of the parallelepiped determined by the vectors , and .

Find the volume of the parallelepiped with adjacent edges , and .

Find the volume of the parallelepiped with adjacent edges , and .

Use the scalar triple product to verify that the vectors , and are coplanar.

Use the scalar triple product to determine whether the points A(1, 3, 2), B(3, -1, 6), C(5, 2, 0), and D(3, 6, -4) lie in the same plane.

A bicycle pedal is pushed by a foot with a 60-N force as shown in the figure. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about P.

(a) A horizontal force of 20 lb is applied to the handle of a gearshift lever as shown in the figure. Find the magnitude of the torque about the pivot point . b) Find the magnitude of the torque about if the same force is applied at the elbow of the lever.

A wrench 30 cm long lies along the positive -axis and grips a bolt at the origin. A force is applied in the direction ? 0 ,3, 24 ?at the end of the wrench. Find the magnitude of the force needed to supply 100 N?m of torque to the bolt.

Let and let be a vector with length 3 that starts at the origin and rotates in the -plane. Find the maximum and minimum values of the length of the vector . In what direction does point?

If \(a \cdot b=\sqrt{3}\) and \(a \times b=\langle 1, \ 2, \ 2\rangle\), find the angle between a and b.

(a) Find all vectors v such that ?1, 2, 1? x v = ?3, 1, -5? (b) Explain why there is no vector v such that ?1, 2, 1? x v = ?3, 1, 5?

Distance from a Point to a Line Let P be a point not on the line L that passes through the points Q and R. (a) Show that the distance d from the point P to the line L is (b) Use the formula in part (a) to find the distance from the point P(1, 1, 1) to the line through Q(0, 6, 8) and R(-1, 4, 7).

Distance from a Point to a Plane Let P be a point not on the plane that passes through the points Q, R, and S. (a) Show that the distance d from P to the plane is (b) Use the formula in part (a) to find the distance from the point P(2, 1, 4) to the plane through the points Q(1, 0, 0), R(0, 2, 0), and S(0, 0, 3).

Show that | a x |2 = | a |2 | b |2 - ( a . b )2.

If \(a+b+c=0\), show that \(a \times b=b \times c=c \times a\)

Prove that ( a - b ) x ( a + b ) = 2 ( a x b ).

Prove Property 6 of cross products, that is, a x (b x c) = (a . c)b - (a . b)c

Use Exercise 50 to prove that a x (b x c) + b x (c x a) + c x (a x b) = 0

Prove that

Suppose that \(\mathbf{a} \neq \mathbf{0}\). (a) If \(\mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}\), does it follow that \(\mathbf{b}=\mathbf{c}\)? (b) If \(\mathbf{a} \times \mathbf{b}=\mathbf{a} \times \mathbf{c}\), does it follow that \(\mathbf{b}=\mathbf{c}\)? (c) If \(\mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}\) and \(\mathbf{a} \times \mathbf{b}=\mathbf{a} \times \mathbf{c}\), does it follow that \(\mathbf{b}=\mathbf{c}\)?

If v1, v2, and v3 are noncoplanar vectors, let (These vectors occur in the study of crystallography. Vectors of the form n1v1 + n2v2 + n3v3, where each ni is an integer, form a lattice for a crystal. Vectors written similarly in terms of k1, k2, and k3 form the reciprocal lattice.) 1. Show that ki is perpendicular to vj if i ? j. 2. Show that ki . vi = 1 for i = 1, 2, 3. 3. Show that k1 . (k2 x k3) =