Given the vectors a = 6i − j, b = 2i + 5j, c = −i − 4j. Calculate a + 2b − c, and R = a + b − c.

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Given the vectors: a = 6i − j; b = 2i + 5j; c = −i − 4j i) Calculate a + 2b − c Step 1: Calculate 2b. 2b = 2(2i + 5j) = 4i + 10j Step 2: Substitute the vectors into the expression a + 2b − c. a + 2b − c = (6i − j) + (4i + 10j) − (−i − 4j) Step 3: Group the i components and j components. a + 2b − c = (6 + 4 − (−1))i + (−1 + 10 − (−4))j; a + 2b − c = (6 + 4 + 1)i + (−1 + 10 + 4)j; a + 2b − c = 11i + 13j The result is 11i + 13j. For parts (ii), (iii), (iv), and (v), we first need to calculate the vector R = a + b − c. Step 1: Calculate R = a + b − c. R = (6i − j) + (2i + 5j) − (−i − 4j) Step 2: Group the i components and j components for R. R = (6 + 2 − (−1))i + (−1 + 5 − (−4))j; R = (6 + 2 + 1)i + (−1 + 5 + 4)j; R = 9i + 8j iii) Magnitude of a + b − c Step 1: Calculate the magnitude of R = 9i + 8j. The magnitude of a vector xi + yj is √(x² + y²). |R| = √(9² + 8²); |R| = √(81 + 64); |R| = √(145) The magnitude is √(145). ii) Unit vector in the direction of a + b − c Step 1: Calculate the unit vector R using the formula R = R|R|. R = 9i + 8j√(145); R = (9)/(√(145))i + (8)/(√(145))j The unit vector is (9)/(√(145))i + (8)/(√(145))j. iv) Direction of a + b − c Step 1: Find the angle θ that the vector R = 9i + 8j makes with the positive x-axis. tan θ = y-componentx-component = 8/9; θ = arctan(8/9) θ ≈ 41.63° The direction is arctan(8/9) or approximately 41.63° from the positive x-axis. v) Direction cosine of a + b − c Step 1: The direction cosines for a 2D vector xi + yj are cos α = x|R| and cos β = y|R|, where α is the angle with the x-axis and β is the angle with the y-axis. cos α = (9)/(√(145)); cos β = (8)/(√(145)) The direction cosines are (9)/(√(145)) and (8)/(√(145)).

Given the vectors a = 6i − j, b = 2i + 5j, c = -i - 4j. Calculate a + 2b - c, and R = a + b − c.

Step-by-step solution

Part i· 5 steps

Calculate $2\mathbf{b}$ .

Substitute the vectors into the expression $\mathbf{a} + 2\mathbf{b} - \mathbf{c}$ .

Group the $\mathbf{i}$ components and $\mathbf{j}$ components.

Calculate $\mathbf{R} = \mathbf{a} + \mathbf{b} - \mathbf{c}$ .

Group the $\mathbf{i}$ components and $\mathbf{j}$ components for $\mathbf{R}$ .

Part iii· 1 step

Calculate the magnitude of $\mathbf{R} = 9\mathbf{i} + 8\mathbf{j}$ .

Part ii· 1 step

Calculate the unit vector $\hat{\mathbf{R}}$ using the formula $\hat{\mathbf{R}} = \frac{\mathbf{R}}{|\mathbf{R}|}$ .

Part iv· 1 step

Find the angle $\theta$ that the vector $\mathbf{R} = 9\mathbf{i} + 8\mathbf{j}$ makes with the positive x-axis.

Part v· 1 step

The direction cosines for a 2D vector $x\mathbf{i} + y\mathbf{j}$ are $\cos \alpha = \frac{x}{|\mathbf{R}|}$ and $\cos \beta = \frac{y}{|\mathbf{R}|}$ , where $\alpha$ is the angle with the x-axis and $\beta$ is the angle with the y-axis.

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More Mathematics Questions

Given the vectors a = 6i − j, b = 2i + 5j, c = -i - 4j. Calculate a + 2b - c, and R = a + b − c.

Step-by-step solution

Part i· 5 steps

Calculate 2b2\mathbf{b}2b.

Substitute the vectors into the expression a+2b−c\mathbf{a} + 2\mathbf{b} - \mathbf{c}a+2b−c.

Group the i\mathbf{i}i components and j\mathbf{j}j components.

Calculate R=a+b−c\mathbf{R} = \mathbf{a} + \mathbf{b} - \mathbf{c}R=a+b−c.

Group the i\mathbf{i}i components and j\mathbf{j}j components for R\mathbf{R}R.