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Chapter 12: Problem 15

Compute the following cross products. Then make a sketch showing the two vectors and their cross product. $$\mathbf{j} \times \mathbf{k}$$

Short Answer

Expert verified
Question: Compute the cross product of the vectors j and k. Represent the obtained cross product, along with the original vectors, in a sketch. Answer: The cross product of vectors j and k is i. In the sketch, vector j should be drawn along the Y-axis, vector k along the Z-axis, and their cross product, vector i, along the X-axis in a 3D coordinate system.

Step by step solution

01

Compute the cross product

To compute the cross product between vectors j and k, we simply use the standard formula for cross products. Note that we have: $$\mathbf{i} \equiv (1,0,0)$$ $$\mathbf{j} \equiv (0,1,0)$$ $$\mathbf{k} \equiv (0,0,1)$$ Now the cross product between two 3D vectors (a,b,c) and (d,e,f) is given by: $$(a,b,c) \times (d,e,f) = (bf - ce, cd - af, ae - bd)$$ Applying this formula to the vectors \(\mathbf{j}\) and \(\mathbf{k}\), we get: $$(0,1,0) \times (0,0,1) = (1 \cdot 0 - 0 \cdot 0, 0 \cdot 1 - 0 \cdot 0, 0 \cdot 0 - 1 \cdot 0) = (0,0,0) - (0,1,0)$$ So, the cross product \(\mathbf{j} \times \mathbf{k} = \mathbf{i}\).
02

Make a sketch

To create a sketch, draw the 3D coordinate system with mutually perpendicular axes: X-axis, Y-axis, and Z-axis, representing the i, j, and k vectors respectively. Draw the vector \(\mathbf{j}\) along the Y-axis starting from the origin, and vector \(\mathbf{k}\) along the Z-axis starting from the origin. The cross product, vector \(\mathbf{i}\), will lie on the X-axis. The final sketch should clearly show the two vectors \(\mathbf{j}\) and \(\mathbf{k}\), as well as their cross product, vector \(\mathbf{i}\), in the 3D coordinate system.

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Most popular questions from this chapter

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle 4,3,0\rangle, \mathbf{v}=\langle 1,1,1\rangle\)

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Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle\), and \(\mathbf{w}=\) \(\left\langle w_{1}, w_{2}, w_{3}\right\rangle\). Let \(c\) be a scalar. Prove the following vector properties. \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\)

A baseball leaves the hand of a pitcher 6 vertical feet above home plate and \(60 \mathrm{ft}\) from home plate. Assume that the coordinate axes are oriented as shown in the figure. a. In the absence of all forces except gravity, assume that a pitch is thrown with an initial velocity of \(\langle 130,0,-3\rangle \mathrm{ft} / \mathrm{s}\) (about \(90 \mathrm{mi} / \mathrm{hr}\) ). How far above the ground is the ball when it crosses home plate and how long does it take for the pitch to arrive? b. What vertical velocity component should the pitcher use so that the pitch crosses home plate exactly \(3 \mathrm{ft}\) above the ground? c. A simple model to describe the curve of a baseball assumes that the spin of the ball produces a constant sideways acceleration (in the \(y\) -direction) of \(c \mathrm{ft} / \mathrm{s}^{2}\). Assume a pitcher throws a curve ball with \(c=8 \mathrm{ft} / \mathrm{s}^{2}\) (one-fourth the acceleration of gravity). How far does the ball move in the \(y\) -direction by the time it reaches home plate, assuming an initial velocity of (130,0,-3) ft/s? d. In part (c), does the ball curve more in the first half of its trip to the plate or in the second half? How does this fact affect the batter? e. Suppose the pitcher releases the ball from an initial position of (0,-3,6) with initial velocity \((130,0,-3) .\) What value of the spin parameter \(c\) is needed to put the ball over home plate passing through the point (60,0,3)\(?\)
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