Chapter 10: Q. 18 (page 824)

If u, v and w are three vectors in $ℝ^{3}$ , which of the following products make sense and which do not?
localid="1649346164463" $(a) u \cdot (v \cdot w) (b) u \cdot (v \times w) (c) u \times (v \cdot w) (d) u \times (v \times w)$

Short Answer

Expert verified

$(a) u \cdot (v \cdot w) does not make sense . (b) u \cdot (v \times w) make sense . (c) u \times (v \cdot w) does not make sense . (d) u \times (v \times w) make sense .$

Step by step solution

Step 1. Given Information

If u, v and w are three vectors in $ℝ^{3}$ , which of the following products make sense and which do not?

$(a) u \cdot (v \cdot w) (b) u \cdot (v \times w) (c) u \times (v \cdot w) (d) u \times (v \times w)$

Part (a) Step 1. The first product is u·(v·w)

The combination $u \cdot (v \cdot w)$ is not defined, since u·v is a scalar and we

need two vectors to form a dot product.

Part (b) Step 1. The first product is u·(v×w)

The product $u \cdot (v \times w)$ is defined and has an important geometrical interpretation. This is examples of triple scalar products.

Part (c) Step 1. The first product is u×(v·w)

The product $u \times (v \cdot w)$ does not make sense because $v \cdot w$ is a scalar which cannot be crossed with a vector.

Part (d) Step 1. The first product is u×(v×w)

The product $u \times (v \times w)$ make sense because $v \times w$ is a vector which is crossed by a vector.

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If u, v and w are three vectors in $ℝ^{3}$ , which of the following products make sense and which do not?
localid="1649346164463" $(a) u \cdot (v \cdot w) (b) u \cdot (v \times w) (c) u \times (v \cdot w) (d) u \times (v \times w)$

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. The first product is u·(v·w)

Part (b) Step 1. The first product is u·(v×w)

Part (c) Step 1. The first product is u×(v·w)

Part (d) Step 1. The first product is u×(v×w)

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If u, v and w are three vectors in ℝ3, which of the following products make sense and which do not?localid="1649346164463" (a)u·(v·w)(b)u·(v×w)(c)u×(v·w)(d)u×(v×w)

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. The first product is u·(v·w)

Part (b) Step 1. The first product is u·(v×w)

Part (c) Step 1. The first product is u×(v·w)

Part (d) Step 1. The first product is u×(v×w)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Geometry

Logic and Functions

Calculus

Study anywhere. Anytime. Across all devices.

If u, v and w are three vectors in $ℝ^{3}$ , which of the following products make sense and which do not?
localid="1649346164463" $(a) u \cdot (v \cdot w) (b) u \cdot (v \times w) (c) u \times (v \cdot w) (d) u \times (v \times w)$