Examples

Example 4: Find a unit vector v with the same direction as the vector u given that u = ⟨7, − 1⟩.

Two special unit vectors, i = ⟨1, 0⟩ and j = ⟨0, 1⟩, can be used to express any vector v = ⟨a, b⟩.

Example 5: Write u = ⟨5, 3⟩ in terms of the i and j unit vectors (Figure 5 ).

Figure 5
Drawing for Example 5.

Vectors exhibit algebraic properties similar to those of real numbers (Table 1).

Example 6: Find 4 u + 5 v if u = 7 i − 3 j and v = −2 i + 5 j.

Given two vectors, u = ⟨ a, b⟩ = a i + b j and v = ⟨c, d⟩ = c i + d j, the dot product, written as u· v, is the scalar quantity u ˙ v = ac + bd. If u, v, and w are vectors and q is a real number, then dot products exhibit the following properties:

The last property, u ˙ v = | u| | v| cos α, can be used to find the angle between the two nonzero vectors u and v. If two vectors are perpendicular to each other and form a 90° angle, they are said to be orthogonal. Because cos 90° = 0, the dot product of any two orthogonal vectors is 0.

Example 7: Given that u = ⟨ 5, −3⟩ and v = ⟨6, 10⟩, show that u and v are orthogonal by demonstrating that the dot product of u and v is equal to zero.

Example 8: What is the angle between u = ⟨5, −2⟩ and v = ⟨6, 11⟩?

An object is said to be in a state of static equilibrium if all the force vectors acting on the object add up to zero.

Example 9: A tightrope walker weighing 150 pounds is standing closer to one end of the rope than the other. The shorter length of rope deflects 5° from the horizontal. The longer length of rope deflects 3°. What is the tension on each part of the rope?

Draw a force diagram with all three force vectors in standard position (Figure 6).