1.2.E: Angles and The Dot Product (Exercises)

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Exercise \(\PageIndex{1}\)

Let \(\mathbf{x}=(3,-2), \mathbf{y}=(-2,5),\) and \(\mathbf{z}=(4,1) .\) Compute each of the following.
(a) \(\mathbf{x} \cdot \mathbf{y}\)
(b) \(2 \mathbf{x} \cdot \mathbf{y}\)
(c) \(\mathbf{x} \cdot(3 \mathbf{y}-\mathbf{z})\)
(d) \(-\mathbf{z} \cdot(\mathbf{x}+5 \mathbf{y})\)

Answer

(a) -16

(b) -32

(d) 5

Exercise \(\PageIndex{2}\)

Let \(\mathbf{x}=(3,-2,1), \mathbf{y}=(-2,3,5),\) and \(\mathbf{z}=(-1,4,1) .\) Compute each of the following.
(a) \(\mathbf{x} \cdot \mathbf{y}\)
(b) \(2 \mathbf{x} \cdot \mathbf{y}\)
(c) \(\mathbf{x} \cdot(3 \mathbf{y}-\mathbf{z})\)
(d) \(-\mathbf{z} \cdot(\mathbf{x}+5 \mathbf{y})\)

Exercise \(\PageIndex{3}\)

Let \(\mathbf{x}=(3,-2,1,2), \mathbf{y}=(-2,3,4,-5),\) and \(\mathbf{z}=(-1,4,1,-2) .\) Compute each of the following.
(a) \(\mathbf{x} \cdot \mathbf{y}\)
(b) \(2 \mathbf{x} \cdot \mathbf{y}\)
(c) \(\mathbf{x} \cdot(3 \mathbf{y}-\mathbf{z})\)
(d) \(-\mathbf{z} \cdot(\mathbf{x}+5 \mathbf{y})\)

Answer

(a) -18

(b) -36

(d) -126

Exercise \(\PageIndex{4}\)

Find the angles between the following pairs of vectors. First find your answers in radians and then convert to degrees.
(a) \(\mathbf{x}=(1,2), \mathbf{y}=(2,1)\)
(b) \(\mathbf{z}=(3,1), \mathbf{w}=(-3,1)\)
(c) \(\mathbf{x}=(1,1,1), \mathbf{y}=(-1,1,-1)\)
(d) \(\mathbf{y}=(-1,2,4), \mathbf{z}=(2,3,-1)\)
(e) \(\mathbf{x}=(1,2,1,2), \mathbf{y}=(2,1,2,1)\)
(f) \(\mathbf{x}=(1,2,3,4,5), \mathbf{z}=(5,4,3,2,1)\)

Answer

(a) 0.6435 radians, or \(36.87^{\circ}\)

(e) 0.6435 radians, or \(36.87^{\circ}\)

Exercise \(\PageIndex{5}\)

The three points \((2,1),(1,2),\) and \((-2,1)\) determine a triangle in \(\mathbb{R}^{2} .\) Find the measure of its three angles and verify that their sum is \(\pi\).

Answer: The angle at vertex (−2,1) is 0.3218 radians, at vertex (1,2) is 2.0344 radians, and at vertex (2,1) is \(\frac{\pi}{4}\) radians.

Exercise \(\PageIndex{6}\)

Given three points \(\mathbf{p}, \mathbf{q},\) and \(\mathbf{r}\) in \(\mathbb{R}^{n},\) the vectors \(\mathbf{q}-\mathbf{p}, \mathbf{r}-\mathbf{p},\) and \(\mathbf{q}-\mathbf{r}\) describe the sides of the triangle with vertices at \(\mathbf{p}, \mathbf{q},\) and \(\mathbf{r} .\) For each of the following, find the measure of the three angles of the triangle with vertices at the given points.
(a) \(\mathbf{p}=(1,2,1), \mathbf{q}=(-1,-1,2), \mathbf{r}=(-1,3,-1)\)
(b) \(\mathbf{p}=(1,2,1,1), \mathbf{q}=(-1,-1,2,3), \mathbf{r}=(-1,3,-1,2)\)

Exercise \(\PageIndex{7}\)

For each of the following, find the angles between the given vector and the coordinate axes.
(a) \(\mathbf{x}=(-2,3)\)
(b) \(\mathbf{w}=(-1,2,1)\)
(c) \(\mathbf{y}=(2,3,1,-1)\)
(d) \(\mathbf{x}=(1,2,3,4,5)\)

Answer

(a) 2.1588 radians; 0.5880 radians

(b) 1.9913 radians; 0.6155 radians; 1.1503 radians

(d) 1.4355 radians; 1.2977 radians; 1.1543 radians; 1.011 radians; 0.8309 radians

Exercise \(\PageIndex{8}\)

For each of the following, find the coordinate of \(\mathbf{x}\) in the direction of \(\mathbf{y}\) and the projection \(\mathbf{w}\) of \(\mathbf{x}\) onto \(\mathbf{y} .\) In each case verify that \(\mathbf{y} \perp(\mathbf{x}-\mathbf{w})\).
(a) \(\mathbf{x}=(-2,4), \mathbf{y}=(4,1)\)
(b) \(\mathbf{x}=(4,1,4), \mathbf{y}=(-1,3,1)\)
(c) \(\mathbf{x}=(-4,-3,1), \mathbf{y}=(1,-1,6)\)
(d) \(\mathbf{x}=(1,2,4,-1), \mathbf{y}=(2,-1,2,3)\)

Answer

(a) Coordinate: \(-\frac{4}{\sqrt{17}}\); Projection: \(\left(-\frac{16}{17},-\frac{4}{17}\right)\)

(b) Coordinate: \(\frac{3}{\sqrt{11}}\); Projection: \(\left(-\frac{3}{11}, \frac{9}{11}, \frac{3}{11}\right)\)

(d) Coordinate: \(\frac{5}{3 \sqrt{2}}\); Projection: \(\left(\frac{5}{9},-\frac{5}{18}, \frac{5}{9}, \frac{5}{6}\right)\)

Exercise \(\PageIndex{9}\)

Verify properties (1.2.5) through (1.2.11) of the dot product.

Exercise \(\PageIndex{10}\)

If \(\mathbf{w}\) is the projection of \(\mathbf{x}\) onto \(\mathbf{y},\) verify that \(\mathbf{y}\) is orthogonal to \(\mathbf{x}-\mathbf{w}\).

Exercise \(\PageIndex{11}\)

Write \(\mathbf{x}=(1,2,-3)\) as the sum of two vectors, one parallel to \(\mathbf{y}=(2,3,1)\) and the other orthogonal to \(\mathbf{y}\).

Answer: \(\mathbf{x}=\left(\frac{5}{7}, \frac{15}{14}, \frac{5}{14}\right)+\left(\frac{2}{7}, \frac{13}{14},-\frac{47}{14}\right)\)

Exercise \(\PageIndex{12}\)

Suppose \(\mathbf{x}\) is a vector with the property that \(\mathbf{x} \cdot \mathbf{y}=0\) for all vectors \(\mathbf{y}\) in \(\mathbb{R}^{n}, \mathbf{y} \neq \mathbf{x}\). Show that it follows that \(\mathbf{x}=0\).