# Chapter 8 - Polar Coordinates; Vectors - Section 8.7 The Cross Product - 8.7 Assess Your Understanding - Page 652: 42

$-10i -5j +5k$

#### Work Step by Step

A vector that will be orthogonal to two vectors let us say $v$ and $w$ is their cross -product. Suppose that the two vectors can be represented as: $v=v_1i+v_2j+v_3k$ and $w=w_1i+w_2j+w_3k$, then their cross product of such vectors can be obtained in the form of determinate as : $v \times w=\begin{vmatrix} i & j & k \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\ \end{vmatrix}=(v_2w_3-v_3w_2)i-(v_1w_3-v_3w_1)j+(v_1w_2-v_2w_1)k$ Here,we have the cross product of two given vectors as : $u \times w =\begin{vmatrix} i & j & k \\ 2 & -3 & 1 \\ 1 & 1 & 3 \\ \end{vmatrix}\\=[(-3)(3) -(1)(1)] i -j [(2)(3)-(1)(1)]+k [(2)(1) -(-3)(1)] \\=-10i -5j +5k$

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