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Chapter 14: Problem 13

Given that \(p=2 i+2 j\) and \(q=7 k\) find \(p \cdot q\) and interpret this result geometrically.

Short Answer

Expert verified
Based on the dot product result, the geometric interpretation of the relationship between vectors p and q is that they are orthogonal (perpendicular) to each other, forming a 90-degree angle in the geometric space.

Step by step solution

01

Write down the given vectors

We are given two vectors p and q: \(p = 2 i + 2 j\) and \(q = 7 k\).
02

Compute the dot product

To compute the dot product of vectors p and q, we multiply their corresponding components and add them up: \(p \cdot q = (2 i + 2 j) \cdot (7 k) = 2 \cdot 0 + 2 \cdot 0 + 0 \cdot 7 = 0\)
03

Interpret the result geometrically

Since the dot product \(p \cdot q = 0\), this implies that the angle between vectors p and q is 90 degrees (since the dot product \(a \cdot b = |a||b|\cos(\theta)\) , and if \(\theta=90^\circ\), then \(\cos(90^\circ)=0\)). Therefore, the given vectors p and q are orthogonal (perpendicular) to each other in the geometric space.

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Most popular questions from this chapter

State the position vectors of the points with coordinates \((9,1,-1)\) and \((-4,0,4)\).

Explain the distinction between a position vector and a more general or free vector.

If the triple scalar product \((\boldsymbol{a} \times \boldsymbol{b}) \cdot \boldsymbol{c}\) is equal to zero, then (i) \(a=0\), or \(b=0\), or \(c=0\) or (ii) two of the vectors are parallel, or (iii) the three vectors lie in the same plane (they are said to be coplanar). Show that the vectors $$ 2 \boldsymbol{i}-\boldsymbol{j}+\boldsymbol{k}, 3 \boldsymbol{i}-4 \boldsymbol{j}+5 \boldsymbol{k}, \boldsymbol{i}+2 \boldsymbol{j}-3 \boldsymbol{k} $$ are coplanar.

Point P has coordinates \((7,8)\). Point \(Q\) has coordinates \((-2,4)\) (a) Draw a sketch showing \(\mathrm{P}\) and \(\mathrm{Q}\). (b) State the position vectors of \(\mathrm{P}\) and \(\mathrm{Q}\). (c) Find an expression for \(\overrightarrow{P Q}\). (d) Find \(|\overrightarrow{\mathrm{PQ}}|\).

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