We read your photo as:

For each of the following vectors, find: a) The magnitude of the vector, b) The unit vector in the direction of the vector i. 6i + 8j ii. -3i + 5j

Mathematics

ScanSolve AI Answer

Step-by-step solution

10 steps

Here's the solution for question 4.

4. For each of the following vectors, find:

a) The magnitude of the vector

b) The unit vector in the direction of the vector

i. 6i+8j6\mathbf{i} + 8\mathbf{j}

a) The magnitude of the vector

  1. Let the vector be v=6i+8j\mathbf{v} = 6\mathbf{i} + 8\mathbf{j}.

    The magnitude of a vector xi+yjx\mathbf{i} + y\mathbf{j} is given by ∣v∣=x2+y2|\mathbf{v}| = \sqrt{x^2 + y^2}.

    ∣v∣=(6)2+(8)2|\mathbf{v}| = \sqrt{(6)^2 + (8)^2}

  2. Calculate the squares and sum them.

    ∣v∣=36+64=100|\mathbf{v}| = \sqrt{36 + 64} = \sqrt{100}

  3. Find the square root.

    ∣v∣=10|\mathbf{v}| = 10

    The magnitude of the vector is 10\boxed{\text{10}}.

    b) The unit vector in the direction of the vector

  4. The unit vector v^\hat{\mathbf{v}} in the direction of v\mathbf{v} is given by v^=v∣v∣\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}.

    v^=6i+8j10\hat{\mathbf{v}} = \frac{6\mathbf{i} + 8\mathbf{j}}{10}

  5. Divide each component by the magnitude.

    v^=610i+810j\hat{\mathbf{v}} = \frac{6}{10}\mathbf{i} + \frac{8}{10}\mathbf{j}

  6. Simplify the fractions.

    v^=35i+45j\hat{\mathbf{v}} = \frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j}

    The unit vector is \boxed{\text{\frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j}}}.

    ii. −3i+5j-3\mathbf{i} + 5\mathbf{j}

    a) The magnitude of the vector

  7. Let the vector be u=−3i+5j\mathbf{u} = -3\mathbf{i} + 5\mathbf{j}.

    The magnitude is given by ∣u∣=x2+y2|\mathbf{u}| = \sqrt{x^2 + y^2}.

    ∣u∣=(−3)2+(5)2|\mathbf{u}| = \sqrt{(-3)^2 + (5)^2}

  8. Calculate the squares and sum them.

    ∣u∣=9+25=34|\mathbf{u}| = \sqrt{9 + 25} = \sqrt{34}

    The magnitude of the vector is \boxed{\text{\sqrt{34}}}.

    b) The unit vector in the direction of the vector

  9. The unit vector u^\hat{\mathbf{u}} in the direction of u\mathbf{u} is given by u^=u∣u∣\hat{\mathbf{u}} = \frac{\mathbf{u}}{|\mathbf{u}|}.

    u^=−3i+5j34\hat{\mathbf{u}} = \frac{-3\mathbf{i} + 5\mathbf{j}}{\sqrt{34}}

  10. Divide each component by the magnitude.

    u^=−334i+534j\hat{\mathbf{u}} = -\frac{3}{\sqrt{34}}\mathbf{i} + \frac{5}{\sqrt{34}}\mathbf{j}

    The unit vector is \boxed{\text{-\frac{3}{\sqrt{34}}\mathbf{i} + \frac{5}{\sqrt{34}}\mathbf{j}}}.

Was this helpful?

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Mathematics Questions

QFind a magnitude and the unit vector of the following given that: a) a = 3i − 4j, b) b = i + 2j − k…QGiven vectors v1 = i + 4j + 2k, v2 = 3i − 3j − 2k, and v3 = −2i + 2j + 6k, find the magnitude of…QGiven vectors a = 2i − 3j + k and b = −3i + j − 2k, find the magnitude of 2a − 3b.QThree vectors a = 3i+4j, b = 6i+8j, c = −6i−8c. Identify which are (i) equal (ii) parallel (iii)…QGiven the vectors a = 6i − j, b = 2i + 5j, c = −i − 4j. Calculate a + 2b − c, and R = a + b − c.
Browse by topic:Linear Algebra
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp